Is this Tamworth BMX? I didn’t realise it was open yet. Must pop down and have a look.
She could probably get away with 20″ wheels rather than an 18″ as she is only going to get taller, just make sure you get one with a short TT. There are loads of BMXs at that price as long as your not a brand snob. As fas as what to look for have a look at what material they are made from, you want full cro-mo or even alu, you are likely to find Hi-tensile steel. Have a look for the size, as you can imagine they dont go up in ST like MTBs they go on TT length with 20″ long being probably the most common.
She will only need a rear brake and for pure track riding she wont need nobbly tyres as they will slow her down.
At perry barr full face helmets, trousers and knee pads (or shorts with knee and shin pads) are mandatory. Dont know what other clubs are like.