Learn English – Have forgotten or forgot

Grammatically, either version is fine for OP's context, but most people would probably go for the shorter Simple Past version most of the time (if only because it is shorter and simpler).

The difference is really one of fine nuance. Present Perfect focuses more explicitly on your current state - that of being without your wallet due to (past) forgetfulness. Simple Past actually carries the same meaning in OP's exact context, but it's important to note that this is a contextual implication.

Consider a slightly different scenario where the speaker is paying his drinks bill with a credit card, but forgetfulness strikes in the form of a senior moment at the "Enter PIN number" stage. Credible things he might say to the barman to explain the problem include...

1) I have (more commonly, I've) forgotten my PIN
2) I forgot my PIN
3) I forget my PIN

I think in that situation the Present Perfect would probably be most likely. Note that in practice the Simple Present (#3) doesn't actually occur very often with the specific verb to forget. We don't tend to think of forgetting as a "continuous" activity - it's usually the case that at some unspecified point in the (usually, very recent) past the speaker realised he couldn't recall some relevant information because it had already been forgotten. Thus to some people it might sound a little odd to use Present Tense to say you're currently forgetting something - which almost implies that immediately prior to speaking, you hadn't yet forgotten it.

Short Answer:

Both are perfectly natural, and it would be stretching a point to say either is "better" in most contexts. But as ever - for simplicity, most learners would do better to avoid Perfect forms unless they're absolutely necessary (the KISS principle).