Let's break this down:
Gorum's Swordsmanship does allow a Chaotic Neutral barbarian or fighter who worships Gorum to apply the effect of the Vital Strike feat to an attack [he makes] with a greatsword at the end of a charge.
Martial Versatility allows [that barbarian or fighter] to use [Gorum's Swordsmanship] with any weapon within the same weapon group--in this case, the heavy blades group.
Lances only exist in the polearms and spears weapon groups, however, so as things stand the character in question could not use Gorum's Swordsmanship with a lance, and would only deal double damage with a greatsword in conjunction with the Spirited Charge feat.
Vital Strike allows you to roll the weapon's damage dice for the attack twice, but the Damage section in the PRPG Core Rulebook (pg. 179) tells us that "Extra damage dice over and above a weapon’s normal damage are never multiplied."
With all that in mind, a Chaotic Neutral barbarian or fighter who worships Gorum and has all the above feats could make a mounted charge with a greatsword and deal 4d6 damage (2d6 from his greatsword multiplied two times, as per Spirited Charge) plus all the appropriate bonuses. He would then add 2d6 damage from Vital Strike (but would not multiply it).
In order for the character in question to get the full benefit of the Spirited Charge feat, he would need the to would need to add the Versatile Design Weapon Modification to his greatsword*, so that it would be considered to be a weapon of either the polearms or spears weapon groups. Then he would have to take the Exotic Weapon Proficiency (or have another means of gaining said proficiency) to use said modified weapon without a penalty.
If he did all this, the character in question could make a mounted charge with said modified greatsword and deal 6d6 damage (2d6 from his greatsword multiplied three times) plus all the appropriate bonuses. He would then add 2d6 damage from Vital Strike (but would not multiply it).
* Of course you could instead add that modification to a lance so that it would be considered a weapon of the heavy blades weapon group, but you end doing 5d8 damage versus 8d6 damage (22 average damage vs. 28 average damage). Then there's the versatility consideration, in that it's probably easier for a mounted greatsword wielder to make a Ride DC 5 check and guide his steed with his knees so he can wield his two-handed weapon than for that same character to run around with a lance while dismounted.
You are correct! I was just fixated with the "optional replacement" text. Not the first time, and probably not the last time, either!
Getting full value from Spirited Charge can be feat-intensive, and could get in the way of the other important consideration I think you're alluding to: reach--which a lance and fauchard give you, but a greatsword doesn't. So if you stick with greatsword, you probably want the Lunge feat. But if you do go the weapon modification route, you probably also want something that allows you to dispense with reach (when someone gets inside your threatened area), like Choke Up from the Polearms section of the Weapon Trick feat.