Hi Von Smyth - welcome to TC!
If you use the search function here, you'll find a TON of tips for seeing all there is to see, Tiki-wise, in the windy city.
A short list of what there is to see...
In terms of a hotel, I'd find something reasonably priced downtown if possible. That being said, I lived there for 16 years, so I never had much need for a hotel - I'm sure others can chime in with recommendations. Lots of great old places like the Drake up to more modern, stylish Kimpton hotels or the W or Radisson Blue in Aqua Tower will give you the "downtown experience." The CATT (Chicago Area Tiki Tour) uses the Aloft hotel by the airport. It's nice and reasonably priced, but you aren't really "in Chicago," you're in Rosemont by the airport.
In terms of being centrally located, and using public transportation in town, that is, unfortunately, next to impossible with where everything is in Chicago. Hala Kahiki and the Chef are immediately West of the city, not terribly far from each other (but probably a 20m drive from each other; half an hour or so drive from downtown, I'd say.) Tiki Terrace is out by the airport in Des Plaines, probably half an hour or more if you're driving from downtown, and depending on traffic (Chicago's traffic can be easily as bad as LA, if marginally more predictable.)
The Blue Line runs to downtown from O'Hare; the Orange line takes you downtown from Midway. The Chef and Hala Kahiki are, theoretically, reachable from the CTA but having done each once, I would NOT recommend it... you're looking at close to 2 hours, multiple transfers and a lot of suburban walking (i.e., no sidewalks)
Cabs/Uber in-between each place could run you $30-$45+ per trip, each way, depending on where you're going and your starting point. Downtown to the airport is about $30 in a cab on a good day. We once took a cab from our place in Uptown (far North side) to Chef Shangri-La because it was their New Years Eve hula show and we wanted to not have to worry about drinking/driving (ours and others on the road)... that was about $50 each way. Not something we ever did again (though we did have a really good time.) Trying to cab/uber between every place, you'd damn near be better off hiring a driver for the weekend.
If you are just planning to stay in the city, Three Dots is easily accessible off the red line or the clark bus. Lost Lake is near the Logan Square blue line, which also runs downtown.
Much like touring LA's Tiki bars, Chicago is a very big place... unfortunately everything is pretty gigantically spread out, especially in terms of Tiki. Your best bet, really, is to get a hotel or Air B&B with parking and rent a car. With that, though, you'll really have to pace yourself. The Chef, in particular, makes their drinks at jet fuel strength. Not proud to admit we've had a few white knuckle trips home from there after a few Mai Tais. Three Dots and Lost Lake follow the classic drink formulas, so expect 2-4oz of booze per drink. If you're staying downtown, both of these are easy enough to get to on the el.
September weather in Chicago can swing between ghastly hot and humid, pouring rain, to sweatshirt weather. Fall and Spring here in the midwest are, unfortunately, pretty schizophrenic. I'd check the weather reports when you're packing just before you leave, then plan for it to be wrong!
Hopefully this is helpful - like I say, use the search function here and you will find many, MANY posts about Chicago Tiki, what to plan for, where to go, probably some hotel recommendations, etc. Also in the "events" section, check out the CATT... they've got it pretty well nailed down.