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Also technique play a roles, so here’s what I usually do.
Fit one side of the tyres bead in first.
Inflated inner tube and fit it in, enough so it get inside the tyres.
Installing the other side of the tyres bead; start 180 degrees from valve and work your way up toward the valve both side.
When it start to get tough, deflate the inner tube, repeat, deflate etc.
When it get impossible, go back and check if all the tyres bead is sitting in the middle of the rims as well letting all the air out and push the valve inward to move the inner tube deeper inside the tyres (this prevent it from getting pinched between rims and tyres).
If all bead is in centre of rims, repeat, check and repeat etc.
If the last tiny bit still doesn’t work, then you can use the tyre bead.
Lastly, very very stupid question, if the tyres is 28c and your inner tube is 28-32, then swap for 23-28, better a little small than a little big.
Can you lift the end of the current tape up a bit? Is it thick and/or multi-layered?
If so, then it might be worth a punt on new tape for (I paid £4 for two rims of Velox). I've got a whole roll of that yellow tape - same as Stan's - for when I was trying to get that stupid front wheel to work by converting it to tubeless. You're welcome to a rim's worth if you can be arsed getting it from me. I'd imagine your nearest bike shop would have a roll of Velox you could experiment with. You might be able to use a skinnier version, so long as it covers the rim holes.