An effective way of ending up with flat prints without a NASA sized budget is to make the borders 5mm bigger than you need and then to use watercolourists tape (they use it for shrinking paper by wetting and air-drying it, pretty much the same job in other words) to fix the sponged/squeegeed print to a rigid impervious surface. A glass door, or a window, or a piece of acrylic-sheet, or formica covered chipboard or . . . lots of things work.
Doing this way the paper shrinks slightly as it dries and the fibres dry flat - they are not dried together in a curve than brutalised afterwards, they really are flat. The only curve might come later when the air humidity changes a lot, but even that is less than a squashed-flat print returning to it's previous state.
The disadvantage of the method for you will be needing to transport wet prints home, though you can try soaking and wiping the prints you now have in order to remove the cotton fibres and then taping them up at home.
In no particular order, as it depends on practicality for you, actions are 1) suggest washing the dryer cloth to the darkroom proprietor. 2) at an art-shop, find some high-quality acid-free paper to cover the surface of your print as it goes through the rotary-dryer (quite a bit larger than the print as it may 'move' as it goes through the curves of the rotary drier). 3) at an art-shop, get some watercolourists tape (it is a standard product and is even cheap, don't use parcel tape as it's often very acidic). 4) Re-wash the prints you have and dry them using paper for protection (at the darkroom) and/or the taping method (at home).