I'm wondering if anybody ever thought of blow-in doors similar to the kind used on gas turbines?
Fair enough question; but blow-in doors in a jet inlet only work at low IAS where the static pressure inside is less than on the outside of the inlet. The compressor (or fan) draws a partial vacuum that overcomes spring force of the doors. As IAS builds, the static pressure inside rises to equal and then exceed static pressure outside, and the doors are forced closed.
A radial engine cowl never sees this pressure differential. (Well - The only possible situation I can think of would be if there were a cooling fan in the system to create a partial vacuum...)