Separate the basement room or do it yourself?

Ok, now I understand it better.
Would still not do anything. You can have a tidier storage room and a messy one (the already existing one). But this way it makes more sense – and it really looks like a manageable amount of own work...

We definitely want to separate the room.

So should I do that as a DIY job?

Then I just need to have the electrician install the light switch for the room and the door near the light switch.

Would you leave the underfloor heating for the hallway and the new room together or have separate circuits for each?

If there is underfloor heating there, you have to be careful when you want to screw a profile for the drywall onto the floor. But do you even need underfloor heating in the storage room and the hallway? I would say you can actually do without it. You normally don't heat a storage room.

The underfloor heating is installed; if it were omitted, there would be no credit for it. That is why we will definitely keep it in.

You seem to be up for walling ;)

I would screw a curtain rail to the ceiling and hang a two-part curtain. You still have three masonry walls left for shelves anyway.