So the bimini is on its last breath and to remove the canvas, I would have to remove the backstay, since it goes through a hole in the middle. Yes, I can just cut the old one, but for the new one, I would still have to remove the backstay to mount it.
Will I collapse my mast or rather, will I be able to put back the backstay without needing a pulley?

Every time I haul mine out I have to remove the backstay so the travel lift can pick it up.

Just run your topping lift & halyard to a point each side on the gunwale or somewhere before you loosen the backstay.

Be sure to loosen off the backstay turnbuckle until the toggle pin is loose. Don't knock the pin out whilst under backstay tension or you will deform the toggle jaws when the pin lets go of one side. If the turnbuckle doesn't have enough free play, rig your main halyard to the stern and grind down on a winch to bend the mast to slacken the backstay.

All good answers here, just want to add one.
Some boats have that much rig tension that all of the above ideas will still not allow you to get the pin out, so you may have to loosen your forestay.
Also, pro tip of the week.....
SIDE CUTTERS
Best took you can have for rigging. You can straighten a split pin, then grab the split pin head and use the side cutters as a lever to pull out.
Always straighten split pin before you loosen rig as the clevis pin won't turn.
Before you put a new split pin in (recommended), open the split pin up about 3mm at the tips, this will stop it falling out of the clevis pin while you tighten rig up again. Once rig tight the open split pin.

Oh and split pins only need to be open 15deg total between legs. If there is no need to bend them right around, just don't!!!