If you're keeping the alchemist, maybe go in strong on the alchemy angle?
Bump the inquisitor to full BAB, remove the +attack judgment and otherwise thin the list (how is this game setting handling DR, particularly DR/magic?), see if there are any replacement judgments you can think of. An easy idea might be varied +elemental damage judgments, or ones to defeat DR like silver or cold iron, themed as being alchemical oils the inquisitor applies to their weapons, making the inquisitor more of a beastslaying Van Helsing sort (without the undead hunter part as much). Depending on how you're handling consumables and equipment, they could be able to invest money into expensive vials that would grant emergency uses of judgment, or possibly also double-strength judgment boosts.
Give bards healing salve abilities. Out of combat, everyone should have access to heal skill stuff that makes healing a little faster than normal (unless you really want to keep things at a super slow pace), while bards would have in-combat and emergency healing capabilities. Then, double down on the lore angle, allowing Knowledge skills to grant in-combat bonuses against enemies, with the bard able to do it as a class feature without investment in individual Knowledge skills. Without the cleric and wizard, bards become much more important debuffers. Maybe even consider things like Knowledge (engineering) allowing you to manipulate the environment; the bard (or anyone else with K (engineering), but the bard gets it for free) might be able to fire an arrow at a crumbling pillar, hit just right spot, and create an area of difficult terrain or maybe deal some area damage.
Retaining alchemists also helps tie into the theme, having them as buffers. If you're cutting down magical equipment, temporary stat enhancement bonuses become much more important, and if the alchemist is the only one who can provide stuff like Bull's Strength, their role becomes a lot clearer. You might want to throw them a bone and give all alchemists the infusion discovery for free, otherwise it would sort of become an obligatory non-choice. Alternatively, turn all enhancement bonuses into morale bonuses; then the alchemist, barbarian, and bard are all playing in the same space in terms of buffs, and it relieves the pressure on the alchemist.
A lot of this depends on how you're doing equipment. Personally, I'm a big fan of giving consumables (or effective consumables) as class features to encourage their use. You could give the rogue free alchemical items they've "requisitioned," saying that they only get more if they use the ones they get, completely removing the pressure of managing cash and the feeling that you shouldn't "waste" consumables. Then you'd have the alchemist (bombs, infusions), bard (salves), inquisitor (judgments/oils) and rogue (general items) with these sort of consumables as class features, giving a nonmagical explanation for their abilities while hopefully encouraging other characters to use more consumables in general. This sort of status quo would also allow you to introduce occasional "really magic" consumables for one-off dramatic effects, while making sure they never become ubiquitous (assuming you want that sort of thing at all; if you're going for a much stronger "no magic ever" setting, then you can just not, of course).