Energy per bit ($E_b$) in BPSK with non-unity channel gain
The energy per bit $E_b$ means one thing and one thing only: how many joules is the transmitter spending, on average, per information bit transmitted. Note that the channel has absolutely nothing to do with $E_b$.

However, your question involves the SNR too. In a channel with non-unit gain, you need to decide where you'll measure the SNR.

You can define the SNR at the transmitter: $\text{SNR}=2E_b/N_0$ for BPSK. Obviously, the BER depends on both the SNR and the channel gain.

You can also define the SNR at the matched filter's output: $\text{SNR}=2G^2E_b/N_0$ for BPSK, for a channel with gain $G$. In this case, the BER depends exclusively on the SNR.

When $G$ is random with known probability density, it is common to average the SNR over $G$, in order to obtain the expected BER over many channel realizations.