I've had asthma since I was born, and keep a rescue inhaler with me all the time. Different things trigger asthma for different people, and also the triggers can change with time. Typically as you age your triggers get fewer and you're more aware of what they are so things are less surprising.
Not all gyms are the same: some are closed in shoeboxes with bad circulation and lazy cleaning schedules, others are disinfected multiple times a day (at least the handles and touch-ables), and others are in outdoor environments with plenty of fresh air blowing in.
8% of Olympic athletes have asthma, and you can be sure they spend plenty of time in gyms and various forms of training. Interestingly enough, 8% is also the national (USA) average of asthma sufferers. To me, that means that asthma is no longer a determining factor in athletic performance in all but the most extreme cases (particularly exercise induced asthma).
I've managed to have a fairly athletic life and my asthma inhaler is like my wallet or underwear: I don't travel far without them.
Most gyms will allow you to use their facilities for a few days or weeks for free. Do that, and bring your rescue inhaler with you. Since you mentioned that particulates and specific allergies are triggers for you, this should be a pretty quick way to determine if those items are present in sufficient quantities at any given facility.
When I was running a lot (half marathons primarily) I would almost always use my rescue inhaler for one puff in the beginning of my training runs, two minutes in, and from there I was great. If I told my doctor that I was using my rescue inhaler once a day, she would rightly freak out, but at the same time my lung capacity vastly improved, as did my VO2 max and O2 blood saturation rebound rates.
In all but the rarest of cases, the data and experience shows that asthma is something you deal with but it shouldn't keep you from a single day of training at maximum.