Master of Many Styles and the Unarmed Fighter archetypes were nerfed to only allow feats with the [Style] descriptor, of which only the first in the feat chain is constituted as, which means you have to meet the requirements normally after the first feat. So,
You're still going to need Weapon Finesse, Weapon Focus, and TWF in order to take the second feat, regardless of how you build, which means your suggested ideas aren't that great.
Dipping isn't that bad, you just have to outweigh the pros with the cons, and find dips that mesh extremely well with what you're currently going for.
My suggestion would be to take your first two levels in Unchained Rogue, since you get Weapon Finesse for free, and can take Weapon Focus or some other combat feat as a Rogue Talent (the Style feat chain are all Combat feats), continue as Vivisectionist Alchemist for two levels (stacking the sneak attack dice), then take your third level into Unchained Rogue, which lets you get Dex to Damage with your spear (since the feat gives you the ability to select your spear with Finesse Training). You can just continue with Vivisectionist Alchemist from there.
Suggested feat path:
1. Two Weapon Fighting, Weapon Finesse
2. Weapon Focus (Injection Spear)
3. Spear Dancing Style
5. Spear Dancing Spiral
If you could be Human, the bonus feat would go a long ways here, since you can have this build "online" by 3rd level instead of 5th level, a 2 level differential in the most played part of the game (1st-5th level).
With Human, you can simply do this:
1. Two Weapon Fighting, Weapon Finesse, Weapon Focus (Injection Spear)
2. Spear Dancing Style
3. Spear Dancing Spiral
You can be pure UC-Rogue until 3rd level, getting Dex to Attack and Damage with the Injection Spear (since Finesse Training only calls out a weapon that you can use Weapon Finesse with, which is possible via the Spiral feat), and then split into Vivisectionist Alchemist for stacking Sneak Attack dice while getting your other Alchemist goodies. Granted, you can't take the Weapon Focus talent since you spent a regular feat on it, it's not that powerful of a talent, and that can be solved if Retraining is allowed at your table (which, since feats like Spear Dancing Style are allowed, I imagine Retraining is allowed as well).
Also, because the Spear is a two-handed weapon, that's 1.5x Dexterity on damage rolls, which is pretty huge; not to mention with the option of going TWF with it due to Spear Dancing Style for racking up Sneak Attack, which grants Dexterity to damage all on its own (because it's still the same weapon) it's a very powerful and versatile combination.
**EDIT** Just saw that you need to spend proficiency for the Spear Syringe. That can be solved by taking the Swashbuckler UCRogue archetype, which automatically grants proficiency in a single weapon Martial weapon of your choice, of which the Spear Syringe qualifies for. You lose Trapfinding as a result, but the option to take the Combat Feat talent twice is also a nice addition if you decide the 4th level of UCRogue is worth it.
Oh yeah, I forgot Weapon Focus has a BAB requirement. Stupid...
You could get Spear Dancing Spiral at 4th via the combat trick rogue talent. The weapon training talent (for WF) is separate so you can get them both without some archetype. You'll need to be a half-elf with ancestral arms to get EWP at 1st, it too has a +1 BAB prereq.
The go-to spell for alchemists using syringe spears has always been skinsend to two-shot any creature (well presumably only those with skin) in the game, but you probably don't want to go that crazy...