The Wallwart PSU for the Phono stage is pretty basic and I doubt there is much in the way of regulation inside.
I found a nice project on the HiFi Wigwam Forum that "Killie99" had completed and his project is the basis for this build.
It is built around the Sigma22 dual rail, tracking, linear PSU, designed by AMB Laboratories and uses the board and kit of parts available from Glass Jar Audio. It uses all discrete components and high current MOSFETs as the output "Pass" devices.

This is what the populated board looks like and is 6" x 3.5" in size.
The heat sinks and a transformer, if required, can be purchased as part of the kit.
All that is needed is a case to put it in and the additional hook up wiring, switches and sockets.

This aluminium case is available from Douk Audio MPN 261686305349 and can be found on eBay.
It's slightly bigger than the Phono stage, but looks similar. With a bit of work it can be modified to exactly match the 206x68x200mm dimensions of the Pro-Ject cases. The 8mm thick front panel can be replaced with a 10mm one to match the phono stage and it can be tapped and secured from the inside so the bolt heads are not seen. An on/off  toggle switch that matches those used by Pro-Ject can be purchased from RS Components # 190-0664.

This is my proposed layout into the modified case, it's a tight fit, but I think it's doable.
I've stuck with the Talema potted transformer that "Killie99" used.
The PSU kit is meant for an amplifier project, not all of the outputs are required for this phono stage, so only a single terminal block will be soldered in place. I have gone for 90deg terminal blocks, where the wires are fed in to the top, this saves space and avoids tight turns on the wires. I measured the output at the mini XLR plug attached to the stock Wallwart supply and included the pin out for reference.