Is there a reason why a pack and play wouldn't work? They have different heights and even sell boards that are more like mattresses than the most common pack and play hard bottom.
And they're very portable and would last very long (my 99th percentile 2yo is sleeping in one this week on vacation).
Ours came with a bassinet height and the lower height for older children. No paint issues, and definitely under $160.
SAHM to Magnolia May (09/10) and Luke Russett (04/13) and wife and best friend to my airman.