Your going to need some other feats to make it work out.
Mounted Skirmisher and such.
I feel your not getting that much benefit considering how many feats your going to be feeding to fighting on a mount. But that is just me. It is your character and if you want to give up your feats to Mounted Combat that is up to you. But I personally wouldn't do it.
One of my favorite things to do on an Alchemist is the Tumor Familiar (Pick something with +2 will save or +4 initiative). Then Give it the Protector Archetype. Use the Extract Ablative Barrier and have some damage you take turn into Non-Lethal..then split your damage with your familiar (as shield other). Your familiar has Fast healing 5 so it can take a few shots a round. This gives you effectively a lot more health for when things start focusing on you.
I was going to go into melee on my mount - a sort of clockwork cavalier.
In fact, I was thinking about having a Goblin with four levels of Unchained Rogue, TWF and then levels in this alchemist.
I was also thinking of Boon Companion as a 5th level feat but HeroLab didn't like it. I can't see why not (for the construct mount).
Just ideas, you know :-)