generator transfer switch install question
No, you shouldn't feed load side equipment through the meter enclosure.

As an option, most panel manufacturers make a generator lock out slide that attaches to the panel and doesn't allow the main and generator breaker to be on at the same time. It is a simple piece of stamped steel but you panel dead-front must be modified to attach it.

In this setup, the generator breaker must be the top right or left (depending on the panel manufacturer) breaker.

Return the transfer switch and save some money.

Good luck and be careful!