If you want to boost your BAB, rather than reducing it, one level of Snakebite Striker Brawler is another way into Sneak Attack @ level one. At level 3, take Accomplished Sneak Attacker...
Enigma gets 1 die of sneak attack at 5th level. Take the accomplished sneak attacker feat at 5th level and you're set. Mesmerist isn't arcane casting though, so getting that second level arcane spell handled is going to be a larger hurdle.
The vizier mesmerist archetype does exactly what you're trying to do by grabbing 5 levels of arcane trickster. Unfortunately, it doesn't stack with either of your current archetypes. Essentially all your spells and class abilities appear to come from someone else of your choice.
There's also the "Conceal Spell" feat from ultimate intrigue that may do what you want.
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Also, casting any sort of spell, even without any components, still has observable 'swirly magical' effects, as evidenced by Spellcraft being usable to identify spells being cast, even when they have no verbal, somatic or material components. So, even if you like, you cannot completely hide that you ARE casting spells. I don't see anything in either of your archetypes to change that for the Mesmerist.
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RAW, the only way to conceal spell casting is the Conceal Spell feat, otherwise no amount of bluff or anything will conceal the visual manifestations of the spell being cast.
With your choice of archetypes you would get the sneak attack by level 9, or earlier if you take the Accomplished Sneak Attacker feat...you would need at least 4 levels of an arcane spell casting class like Wizard to reach 2nd level arcane casting though. Taking 4 levels of a different class would basically cripple your Mesmerist casting though...losing 4 character levels of a class which already has diminished casting ability would be super painful. (You would not get your level 6 spell slots until level 19, instead of the normal 16)
Interestingly, the way Arcane Trickster is worded, you could actually increase your Mesmerist caster levels with it (Or any caster class really)...I don't think that was intentional on the part of the writers though.
(sneak attack class) 1/wizard 3/mesmerist 1/arcane trickster, increasing mesmerist casting, seems almost strictly worse than (sneak attack class) 1/wizard 3/mesmerist 1/arcane trickster, increasing wizard casting.
If you're going for stealth-casting mesmerist I'd suggest not using the arcane trickster PrC at all. If you want to add damage there are better ways, and for stealth casting it offers nothing to a psychic spellcaster.
Be a Kobold with the Spellcaster Sneak alternative racial feature.