Thanks @mironc. You were right, the select/start buttons were the wrong way around. I opened up the picade and found I’d got them the wrong way around when connecting to the HAT.
I still had the problem with keys popping up in-game menus though. This post was helpful for me to understand what’s going on.
What’s happened was the menu had popped up and I’d gone into the game settings and selected one of the first options (under the Input (general) menu, and then pressed down thinking it would move to the next item, but it actually saved the down position as the configure menu key. So every time I pressed down in the game, it would show the menu.
What I’ve done to “fix” it is press the hotkey to get the menu up, then went into Input (general), then map Config Menu to return (enter button on right of picade) and On Screen Display to escape on the left side. This was tricky, as this utility keeps adding OR to the mappings. So if I press a key e.g. A, then press enter and try to map another key e.g. B, it maps to A OR B. I have to keep doing it until the maximum number of ORs are reached and it records a single key.
This has made the game playable. Now I just need to override the P1 1-6 buttons to get the right combo for SF2!
All of this tinkering around was totally worth it when I saw the delight in my 8 and 11 year old boy’s faces when they played SF2 for the first time and I taught them the special moves - priceless!
Thanks for your help