Are EC public/private keys significantly weakened by having a known byte?
By fixing the first byte of both the private and the public key you actually reduce the key space by about 16 bits, because only about one in $2^{16}$ key-pairs has that property – as you notice. (If there is no way to exploit the restriction on both keys at the same time, you may lose less security but I would assume the worst.)

Doing this is fine, unless the elliptic curve system is almost too small anyway. If it was insecure, then one in e.g. $2^{16}$ randomly generated keys would be similarly weak.

So worst case you should have ~112 bits of security since a 256-bit elliptic curve usually has about 128 bits of security, but that should be enough.