>>> Assuming it’s just UA5 at fault, is it just a matter of replacing that chip?
Yep. Of course, my MARCH-C memory test may find another fault when it is run next...
>>> Would you socket or solder it back in?
Always use a good-quality socket.
>>> I’ve found MM5290J-2 on eBay but mine are MM5290J-4, can you mix them?
The 'MM' means National Semiconductor.
The '5290' indicates it is a 16K * 1 DRAM device.
The 'J' indicates the packaging type.
The '-4' indicates the speed rating.
A '-2' is faster than a '-4' so is a suitable replacement.
A 'bog standard' 4116 DRAM will do perfectly well, assuming the speed (access time) is fast enough. Check the data sheet(s) for the speed ratings. The '-4' has an access time of 250 ns and a '-2' has an access time of 150 ns. See:
https://minuszerodegrees.net/memory/4116/datasheet_MM5290.pdf.
You can (generally) 'mix and match' DRAM in a PET.
I would think twice about using eBay. Suitable 4116 devices are available from a number of suppliers yet. Although, putting the part in a socket, means that you can simply swap it if the replacement turns out to be faulty. However, this gets annoying...
Dave