Under $1,000 I would probably recommend a Browning BPS pump gun, Hunter model, with a 28" barrel .....good light gun...
We've discussed this before / but I don't like a gun under 7 1/2 lbs ....even in the field - but I do have a Benelli Super Sport 20ga, semi-auto, that is 6.2 lbs so it would meet your criteria / but not your budget. Benelli has other offerings in 20ga - but under $1,000 will be a problem ...
You might find a Ruger O/U 20ga for under $1,000 - or maybe a used Browning Lightning series O/U in 20ga - but they're probably up around 7 1/2 lbs so too heavy ... A used Rem 1100 or 11-87 might meet your needs in the semi-auto category ...