Considering how close the prices are in the 350-600 W range, I would probably just buy something more powerful than a 350W. A 350W would probably work ok though. I have a 2009 Antec Signature 650W that I use for most old things.

An enthusiast shopping back in 2003 would probably have gone with a ~400W PSU.

A good quality 350 Watt power supply should suffice, assuming if you're connecting only a few peripherals.

New old stock power supplies can deliver a higher +5V amperage than 12V, your system draws most power from the+12V rail, so recent model PSUs are fine.

Only systems you need to worry about PSU with are ones that place very heavy demands on the 5V line. P4 introduced the ATX12V connector and drawing power from the 12V line as is still done today. So a modern PSU is fine in terms of balance.

As for total Wattage - even though later P4 CPUs were considered hotheads in their day, the highest TDP of a Netburst desktop CPU is 130W and that's dualcore. SInglecores top out at 115W. Compared to modern CPUs that can turbo over 250W that's child's play, in particular given P4 era GPUs were very low power compared to current cards (the FX5900 was considered notoriously hot at 59W, Even the "low-power" RTX3050 draws twice that. You really don't need more than 300W for a P4 system, the only reason to go for more is because the lowest rated PSUs tend to be low-end crap. Go for a good PSU, but even the lowest power good one will be more than enough.

Thank you guys I will consider all your recommendations, much appreciated 🙏🏻🙏🏻