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Have had this issue, tube type rims, tube type tyre, destroyed tyre so just got whatever local place had (somewhere in north of england) which was a tubeless tyre, 1st time ever I wrecked a tube getting it on as was just the wrong bead for the job.
Heidanau make tube type tyres for loads of uses, don't expect a Michelin ps4/ps5 competitor, more like oldschool classic bike tyres. Life much easier with the right ones if you can get them.
If you like the bike and never really use it offroad, get a wheel builder to swap out rims for something tubeless specific, they don't cost that much to get done (rim is £90-200 for something branded, spokes around £40-50 and labour around £50-75 per wheel is about right I think).
BrickMan-
It's not an off-roader per-se. It's a 40yr old BMW R80 and I've been maintaining it for almost 18 years so I guess I must like something about it :) There were a few other wheel types made for the twin suspension model but the snowflakes are my favourite.
Having seen the setup I might try to go tubeless and find the best way to change to a valve in the rim, either drill the current one or find the specialised smaller valve.
Airhead
Yes, but it's a bit more complicated because the wheels are rated for tubes but the tyres are tubeless. Because nowadays no one makes tyres for tubes. Some people run them without tubes but I've never had any luck with that because the holes are not made for the standard valve and I've not gone as far as drilling them out due to the design. The discussions on this subject have been going on for a long time in airhead circles!
But basically, yes, I tried inflating with a a track pump to around 40 but it needed 60 with the valve out on a compressor and plenty of lube. I will try with the valve out on a track pump and go a bit higher on the pressure when I do the rear later today.
I'm guessing motorcross tyres and rims are built for tubes so have a different shape or I need to do a better job of cleaning the inside of the rim.