Mini splits in kitchens are not a good idea, even if you clean them alot. Sooner that later something WILL break. To much humidity and oil to be practical. Yours is the perfect case though, better roll up your sleeves for that one!
If you're looking for "cheap" up front, then just keep doing what you're doing. I would check to make sure the exhaust fans and make up air are balanced (clean too) first. Most places like that have a design flaw such as no MUA, not enough exhaust, other equipment added after the initial design ect... I had a restaurant that grew too fast and needed a temp fix for additional cooling. Rented a portable 2 ton unit with a ducted condensor to the outside and ducted the return from a conditioned storage space (non greasey air). It took up some needed space, but when the kitchen temps got over 95*F, the space wasn't near as important anymore. Good luck!