"None of the other quotelike operators (q// et al) have suffixes so it doesn't seem regular..."

But qr// already does have other suffixes. See perlop under "Regex Quote Like Operators". qr// allows for qr//imosx. So the reason for not adding /e can't be because other quote-like operators don't have modifiers. The main reason is that it doesn't make sense since /e only causes an eval{} block to wrap around the right-hand portion of a s/// operator, and qr// doesn't construct that right-hand portion in the first place.

tie probably had it; don't call it /e. Call it /E or something else to avoid confusion with the longstanding function of the /e modifier.