Can't answer from experience, but what's wrong with a regular RJ45 plug? Otherwise, I would probably ask one of the connector products whether this worked well with the thin wires. If you don't have or don't want to buy a tool and have a little more space for the connection, I would probably try a plug like this: https://www.digitec.ch/de/s1/product/delock-stecker-rj45-cat6a-netzwerk-zubehoer-netzwerkkabel-netzwerk-zubehoer-12757373. Good luck!

I can't answer from experience either, but I have found the following Delock RJ45 connector for AWG32:
https://www.brack.ch/delock-ste...
All other connectors were only for AWG22 up to a maximum of 28
Unfortunately, the connector is only available in packs of 100, but you may be able to find it elsewhere (shop or electrician) as a single connector.
I also wish you good luck.