To my delight, I found out that K-Meleon supports PKCS11, specifically, the Estonian ID-card (national smart card used by for personal authentication somewhat like a passport), without any problems. Great job!
Installing it for K-Meleon is a bit tricky but not too difficult. It can be done from this page:
by first installing the certificates and then importing the module from C:\Windows\System32 (Mozilla support for ID-card must be installed beforehand).
I have asked the developer of Mozilla support for Estonian ID-card to add autodetection for K-Meleon in the next Mozilla support installer.