Leave the perspex, if you have a crash or go over a particularly 'spiky' bump it could shatter. Which is not ideal.
If you want one 'cheap' then homemade is the way to go. I would second the For Sale sign recommendation. I would go one step further and have it 2 ply, with the 2 pieces perpendicular to one another. Pretty stiff then. Stick your map in a plastic bag and use some elastic bands as your retaining system. Very quick to remove the map and fold it over into other posiitions/views with that system.
How does your map-reading part of your brain work? Are you happy with having the map pointing North all the time or would you like to be able to orientate it for your direction of travel? Personally I like to keep it North up at all times as I find it difficult to keep track of where I am if I am constantly reorientating the board for direction of travel. If you want to be able to orientate it for direction of travel I would suggest you look at one of the Miry or Nordenmark boards which can be pivoted around. The bad thing is that these shot up in price significantly recently – exchange rate based I think.
Nordenmark one is the best. I have that and the quick release system is useful for taking the map board on and off