Out of curiosity, will it be a PF character or 3.5? If the latter (particularly if you are using Gestalt characters), you might want to ask your question in the D&D 3.5/d20/OGL section of the forums instead of the Pathfinder section.
My single biggest piece of advice on this build? Whatever you don, don't lose a single point of BAB. Ever. Once you take Abjurant Champion, your spellcasting is keyed off of your BAB, making full BAB immensely powerful, and since this is partial gestalt, you have no need to lose any BAB at all. Even Wiz3//Swash3>Swash2/AC5 works.
Now, then. You may want to start out with Swashbuckler3//Monk1/Wizard1/Factotum1 with Kung Fu Genius [Dragon 319/Compendium] or Carmendine Monk [Champions of Valor], advancing to two levels of Warblade, then your five levels of Abjurant Champion. From there, you can go wherever. Returning to Warblade, possibly for the rest of your career, would not be a bad idea. If Tome of Battle is out, even just returning to Swashbuckler isn't bad, since advancing BAB advances your spellcasting as well. Alternately, since you probably want Wizard in there from the get-go, there is the option of Swashbuckler3//Wizard1/Factotum1/Monk1, taking Kung Fu Genius at level 3.
Kung Fu Genius/Carmendine Monk shunt Monk abilities over to intelligence. Factotum lets you spend an inspiration point to add intelligence to an attack roll, damage roll, or save. At least a couple instances of Font of Inspiration become handy so that you can have more than two inspiration points to work with. Warblade's in there because it's a powerful, full-BAB melee class that can add intelligence to various things, but if Tome of Battle's not allowed, pretty much any full-BAB class will do.
Four or more levels levels of Swiftblade make another a candidate after you finish out Abjurant Champion, letting you add intelligence to initiative while maintaining full BAB. Mind the weird Haste requirement, though. Despite the low hit die, it may be worth riding the class out to the end, since it has some powerful abilities.
Keen Intellect [Dragon 318] can be handy if you can get your hands on it; intelligence to will saves and a few skills. You're almost certainly going to have a great dexterity score, so Insightful Reflexes isn't so useful.
The result of all this? For the first nine levels, you're pretty much pure melee, with a spot of buffing (Mage Armor is your friend), but you get to add your intelligence to a lot of stuff; damage, AC, limited additions to attack/saves/damage (again), possibly will saves, possibly even a bit to reflex saves. Your spellcasting will advance a bit as you go through Abjurant Champion, and then spike at the end, taking you straight from CL5 to CL10; make sure you keep on top of your Spellcraft and have a decent spellbook for when you hit that spike. Also, decide early if you want to make the sacrifices necessary to go into Swiftblade. Reserving all of your third-level spell slots for Haste for your entire tenth level is a pretty big deal (though less so since you get access to fourth- and fifth-level spells at the same time).
Do be aware that maintaining combat gear and a decent spellbook at the same time is very expensive. That level of Monk for intelligence to AC helps defer a lot of the expenses that would normally go towards keeping your AC up, as will Abjurant Armor. A Shield spell that grants +9 AC as a swift action is very nice. One thing to note, though; Abjurant Armor says it applies to abjuration spells that grant an Armor or Shield bonus to AC, but it singles out Mage Armor as an example; an abjuration spell. Ask your DM about this, as +5 to your Mage Armor would be a godsend (or better; +5 to your Greater Mage Armor [Spell Compendium]). Note that come level 10, an extended Mage Armor pretty much lasts all day. It's more practical to use a lesser metamagic rod to extend your Mage Armor effects than it is to actually take the Extend Spell feat.
Don't worry about Monk abilities like Unarmed Strike or Flurry of Blows. Just stick to a sword and use the AC boost.
For stats? Your highest priorities are Dexterity and Intelligence, followed by Constitution in a distant third. This makes an elf a good candidate, though human is always a good decision. I wouldn't recommend starting with anything less than 16 Dex, 18 Int, and 14 Con as a human, or 18 Dex, 18 Int, and 12 Con as an elf, if at all possible.
Crap. Ninja'd by the constraints. Well that sucks. Though the feat thing seems rather unreasonable, since some classes don't even exist in core and rely at least in part on non-core feats that go along with 'em to improve their class abilities.
Most of the more involved advice is bunk due to the restrictions, unfortunately. However, again, don't lose any BAB. Ever. If you can talk your DM into one non-core feat, make it Daring Outlaw [Complete Scoundrel]; a couple levels of Rogue in the gestalt phase (with that one Wizard level taking the third) plus a single level in any full BAB class that grants at least one die of sneak attack damage later (Nightsong Enforcer is good) will let your Swashbuckler and Rogue levels stack for determining sneak attack dice. It may be level 11 before you get there, but finishing out your career with Swashbuckler levels that advance both Sneak Attack and spellcasting on top of solid skill points and rogue class skills can be a very good thing.
Wiz3//Swash3>Swash2/AC5 works.
The problem with this is while you maintain caster level you still have 2 levels where your spell advancement is stalled. I would suggest that the 2 levels of advancement lost from taking 2 levels of swashbuckler are not worth the caster level increase. Also, if you go straight wizard on the non-AbChamp/ Gestalt rules then your caster level will be full regardless.
I would suggest just take the first 3 gestalt levels then wizard 4 then Abjurant Champion. Your BAB drops by 1 versus your original design but you don't sacrifice any spell advancement. Being able to cast 6th level spells at 11th level is more than worth the drop in BAB.
Off the top of my head I'm not sure Swashbuckler gives you the weapon proficiencies you need for Eldritch Knight.
The problem with this is while you maintain caster level you still have 2 levels where your spell advancement is stalled. I would suggest that the 2 levels of advancement lost from taking 2 levels of swashbuckler are not worth the caster level increase. Also, if you go straight wizard on the non-AbChamp/ Gestalt rules then your caster level will be full regardless.
However keep in mind, those stalled levels don't mean much; the character's still a fully-functional melee and possibly sneak type until Martial Arcanist rolls around. There's no real need to lean so heavily on magic until then, whereas if you lose the BAB, that's a permanent scar that reduces effectiveness forever. I'd rather be a few levels' casting behind at the start with ample other abilities to fall back on than end up forever two levels behind in casting later, when I want to be able to rely on my spells.
The problem with this is while you maintain caster level you still have 2 levels where your spell advancement is stalled. I would suggest that the 2 levels of advancement lost from taking 2 levels of swashbuckler are not worth the caster level increase. Also, if you go straight wizard on the non-AbChamp/ Gestalt rules then your caster level will be full regardless.However keep in mind, those stalled levels don't mean much; the character's still a fully-functional melee and possibly sneak type until Martial Arcanist rolls around. There's no real need to lean so heavily on magic until then, whereas if you lose the BAB, that's a permanent scar that reduces effectiveness forever. I'd rather be a few levels' casting behind at the start with ample other abilities to fall back on than end up forever two levels behind in casting later, when I want to be able to rely on my spells.
Where are you losing caster level? after you finish Abjurant Champion?
Regardless, losing spell progression is much worse IMO than losing caster level. Caster level gets you a little extra damage and a slightly better change to bypass spell resistance. You can also make up for lost caster levels with Practiced Spellcaster, you can't ever regain lost spell progression.