Remap the key inside your coin door to F2 then... problem solved.
But seriously the problem probably isn't the remapping, it's the way the service button operates. Each and every game is different.... in some games you hold service.... in others you hold service and then restart the game. In others you tap the service key. Some games don't even have a service menu. Also, and here's the one that is probably throwing you off... the key isn't always called "service" every once in a while it's "service0" or "service1" or what have you because there are multiple service buttons for that game. Before you ask it's possible to remap them all, but in some cases you won't be able to navigate about in the service menu because there is both the regular "service" key AND "service0" or what have you.
I can tell you ahead of time that one button is not enough. If were are talking every single game in mame, you need 4 or 5. There's the key to get into the service menu, adjustment keys (up, down) and typically a "enter" or "service 2" key. You'll probably want a reset key as well as most games require a restart. And which one does what varies from game to game. Now most of these keys can do double duty and you can re-map them to your gameplay buttons, but service 2 and reset will need dedicated buttons at least.
Hope that helps, and seriously though, it would be a lot easier just to remap that button to f2, it's one less thing to worry about.