It is likely a Reactor that was factory converted to a Q*Bert. There were no conversion kits for Q*Bert and that was certainly a Gottlieb cabinet.
Also, that is a standard sized cabinet. I have owned all sorts of standard sized cabinets. Yes you can get a standard Streetfighter layout on it. I do not have that specific cabinet myself, but have owned an Elevator Action and an Arabian that were both converted to Streetfighter 2, and I have seen a picture of a Q*Bert converted to one also. So the controls will fit just fine.
You are likely just getting the idea that the controls won't fit from the every large crowd of younger Mamers here who have never really even seen an actual 19" classic cabinet, much less played one. All they know are the showcase cabinets and Mortal Kombat 4 cabinets, which is why you see so many monsters like showcase Jungle Kings, and 4 foot wide control panels hacked onto Centipedes.
Also, measure that cabinet again. No was is it only 19" wide. I own a tiny sub-caberet Space Firebird, that looks like a Dwarf compared to even a standard mini cabinet, and that is 19" wide. You have a full sized (albeit short) cabinet. Maybe the panel is 19" wide, but the cabinet itelf has to be a bit wider than that (the industry standard was 23-24" for a long time).
I have at home a 19"x5" control panel that has on it 2 joysticks, one bank of 6 buttons, a trackball, 2 start buttons to the right of the trackball, and it still has an inch to spare on each side.
Also, if your son is only 9, consider maybe just going with a 4 button layout instead. Less cramped.
Just trying to keep you from making a mistake. My first Mame cabinet was a "monster" like the ones I fight against now. You know where it ended up? In the garbage. The minute you truly hack up a cabinet you have sealed its fate. It will eventually go in the dumpster.