It turns out I did not have a full understanding of the problem and it was not really a problem. The blanking (pin 4) on U5A AD602 doesn't shut down the channel, it just reduced the gain by 100. So if the gain in previous stage is large enough, the lock can still be acquired. And that's what was happening.
Ideally, we need to keep the AD602 on all the time and lock by scanning the offset with low gain. The loop will catch the lock (the exact same thing I thought was a problem) and once that has happened, we can just increase the loop gain the set value.
Presently, the gain behind the U5A AD602 is 101, which is kind of high. I just need to check if the above-mentioned locking method would work robustly without wrongly getting locked to any higher-order modes with the gain slider set to some threshold value within -10 dB to 30 dB. If that can't be done, I might have to reduce the gain in the first stage. For now, the cavities are locked and beatnote is traveling towards set point.