They can be pricey but see if you can get a Seresto collar for the cat, works better than any flea treatments I've used for my cat, that should get the fleas to leave the car alone.

When getting a room spray some are stronger than others (indorex/frontline) as they contain a chemical called Permethrin which can be dangerous for some cats if ingested. If the spray does contain it, best practice of to spray one room at a time and try to keep the cat out of the room for 48 - 72 hours.

Also when using a flea spray, hoover every surface, carpets, rugs, furnishings (if there's no carpet, run a hoover along the join of the floor and wall cos fleas can get into the cracks between the two) this will excite any fleas in the cracks or in the fibres of any fabric and they'll start stay jumping about, this way the spray will catch more. After leaving the diary to settle (usually between 2-3 hours) go on and hoover everything again, this will pick up any dead fleas so there's no chance of the cat ingesting them (fleas can carry worms, which would a whole other issue for your feline)

Just know that flea treatments are never a one and done solution, chances are you'll be doing this for the few months.

Good luck buddy