I would get the Brymen BL21S2-T4SC test leads, whose probe tips have a threaded base for the T4S screw-on banana plug adapters - then you can attach both 2mm and 4mm probe tips.
I think those are basically the ones that I got with the meter - It's a 2mm probe that accepts a thread-on 4mm banana plug that I have.
Great, then they should work with smooth/unsprung 4mm sockets too.
Thanks for the tip on the "spring loaded socket" - I didn't know how they lead connect. So I presume the one I linked to on Ali-express, when it said it has a 4mm connector then it's likely 4mm spring loaded for the larger connector and not designed for the banana plug?
That comment was about the type of socket you need if you only have bare 2mm probe tips with no other attachment.
Looking at the photos for the aliexpress ones more closely, they seem to be threaded and hopefully compatible with your Brymen leads. Worth a punt for $3 anyway.
The Hirschmann ones do look to be what I want. They seem tough to acquire though...
Also a bit pricey and only CAT I rated - but good if you need compact 2mm probes.