Bulk:
After testing out making an alchemist, he had 2 bulk for the required Alchemist's tools (60 sp) , the free but required formula book is 1 bulk, crossbow is 1 bulk (30 sp), bolts are Light, leather armor (15 sp) is 1 Bulk. That 5 bulk and 1 Light object at the end of character creation. That's before making any alchemical items such as bombs or elixirs.
Potential solution: Could be to give the Alchemist a class only Tool set that weighs nothing due to the fluff reasoning of they know how to minimize and store stuff on their person idk. I would also make All alchemical items just 1 Bulk.
RP & Quick Alchemy & Action Economy:
This is kinda meshed with action economy too. You can make some items at the start of the day at the expense of the ability to make them on the spot with QA. I get that and its understandable. My problem with QA is the lost of effect at the end of the round (or next round with lv 4 feat Enduring Alchemy). If you can't manage to use the item that round its a wasted RP. That's fine too until you try to use poison, which in most cases you can't because it takes 3 actions to apply almost all the poisons. So unless you have quick that doesn't work at all. Also handing some items to use fellow players costs them RP to use after you already spent some?
Potential Solution: QA takes 1 or 2 actions and allows you to use whatever item it is you made. So for example you make Black Adder Venom (injury poison) and apply it to your weapon all as one Activation. This still doesn't make contact or ingested poisons viable since they loose effect at the end of your turn. As to the RP, just give Alchemist a class pool like pretty much everyone else. Spell points and ki and channels, its not hard to separate the ability to make alchemical items (which aren't magical) from RP and make it easier.
Money Issues:
The limits on making items and having to constantly fall back on a weapon due to limited RP to make enough bombs for encounters per day isn't bad, but how many bombs do they get per day? And at the cost of their support/utility. Casters get cantrips to fall back on why can't alchemist get a ability of a low 1d4 or 1d6 bomb with splash at will? Also crafting Alchemy items in the down time is a lot of money and don't gain the infused trait. That means they don't get any Alchemist feat benefits for the most part. I could be wrong but that would also mean that the bombs won't scale since they don't benefit from Empower Bombs at 3rd level.
Potential Solution:
This is really just an extension of the RP problems to me. I think they would have the same solution. Giving Alchemists a way to create more items via "spell points" just seems like the easiest way. Otherwise the alchemist will never have enough downtime or money to actually make enough items to carry around so they dont have to constantly rely on making an item via QA in order to meet a demand on the spot. I mean they are alchemist why should it be so hard for them to make stuff in down time?
Mutagens:
I'll be honest I haven't looked at them in the book too closely, but from what I understand they aren't available until level 5 I believe? Then they take 2 rounds after using them to take effect? That seems a bit slow to activate, but maybe that's just me.
Potential Solution:
I don't really have one since I, admittedly, don't know much about them.
I am curious what level the alchemist in your group is and if they have any comments or concerns about the class? Maybe not now but some they see as the character levels up?