Here's my suggestion:

All season ticket holders who didn't request a refund (and who want to go to games) are asked to form groups with whoever they would like to attend games with (groups are 1-6 people). I'm sure it would be possible to do this on the ticketing site. So you'd end up with somewhere between 3-5,000 groups I'd guess.

Groups are drawn at random one by one and everyone in that group gets a ticket to the game. Keep going until 1,000 tickets have been allocated.

Next game where fans are allowed in same process again with those groups that went to the Saints game being excluded. Keep going until everyone had been to a game them start again with all groups back in.

Fairest way I can think of.

Please ignore this poster.

How do you know? We haven't even tried it. Between March & June is understandable,but looking at this graph, tells me we can be smart enough to give a fans return trial. However, i think we'll just have to agree to disagree on this one.