I am going to avoid telling you HOW to nurse your munchkin in public since you have said in your post that baby does "better" on a bottle...
My 3rd munchkin ended up in the hospital at 4 months for a few days to figure out why he wasn't gaining weight... Turns out he was a lazy nurser and would have been happy to take 2 hours to nurse... But as a mom with a 5 yr old and 2 yr old who both nursed for the "usual" 20 - 30 minutes on a slow day I did not know he wasn't getting enough.(he was never fussy and slept thru the night just like his older siblings) So we started pumping and supplementing. After going thru that with him I am def. more nursing centered and will be taking it so much slower with #4 who is currently tying my intestines in knots with her toes...
If I had to go out, I would schedule it so that I could heat a bottle at home and give it to him in the car. But he was older and had the ability to hold it himself. We did have a bottle warmer for in the car... But used it all of 2 times. I was not impressed by it at all... I don't even know what brand it was since it was a hand me down from a friend.
I like the idea of heating water and using an insulated bag to keep it warm. Maybe heat the water and put it in a glass jar or baby bottle with a lid and just drop a bottle of breast milk beside it in the bag. After a while you then also have some warmish water to clean up any sticky kids. :-)
What would the babys reaction be if you latched him/her on to soothe the belly until the bottle is warm enough?
Carrying a regular bottle warmer along with you may also be a good idea. Even if in a store you can find an outlet and be on your way in 5 min or less. :-) If in a restaurant you could ask the waitress to heat it for you... Make sure to tell them it CAN NOT be microwaved.
It will get easier as babys feeding times grow farther apart...Also, use your hubby... I ended up doing most of my grocery shopping at 8pm so that hubby could get the 2 older in bed while I nursed baby then I would go do what I had to do while hubby kept the baby. THen I would feed him again when I got home.
That cut down a lot on the amount of running I had to do with the kids...