Asking for permission or lawyering up are good options. But if you just want to put it out there do this:
First, rename your game. Make it something completely unrelated to what the old one was called.
Second, change the play a bit more and for your sake change the cut scenes and art work so as not to look like the old game. If they used an old west theme you go with art deco.
Third, once you've completed your changes then only copy off the finished game from your computer and then melt the old drives on your computer into aluminum ingots and melted slag. Or use shred: less fun but not so messy.
Fourth never use this account again. Ever. Better yet change your name and move to a different state. If anyone knows you by sight and knows you used this account then consider plastic surgery.
If the owners of the game ever decide to sue you and they have any way to prove that you made your game from their game they will claim derivative rights or some other ass puckering term to get what they can from you.
if you do make inquiries for permission they will want to see your game so they will know what to look for if you decide to release it anyway after they deny your request.
If you must make inquiries for permission then frame your questions as though you are considering making this game and ask what would their response be to a hypothetical game that you are not yet working on: use an alias and go through a law firm so that you may be insulated so that if they say no and you want to do it anyway then they don't have your address and real name and the law firm has the right to claim confidentiality to keep your real contact information hidden.