I'm not sure that for "cheap" we could really improve.
If by cheap you expand it to be one of Green/Bass plus Bradley Plus a Pick, you could expand to these potentially available bigs:
Marc Gasol/Randolph (depending on circumstances, and would need some additional salaries to make it work)
In terms of cheaper, for varying reasons and for varying packages:
Nene would be harder to get than Emeka and we would rather have Emeka because he has only 2 years left at 14M instead of 13M over 3 years like Nene does.
Al Jefferson just makes too much money to be attainable for this team. 15M means we have to trade Green and either Bass or Lee and someone else. He's also not particularly great at playing defense and he's a FA at the end of the season so we could end up with nothing if we don't wanna pay him a lot. I also think Millsap is a better investment personally.
Varejao is an interesting case. Cleveland can probably get more for him than the Bass-centered package we'd be offering but if they would take some combination of Bass/Melo/pick then I'd do it. I think they might too.
Marc Gasol isn't going anywhere. The Grizzlies would be absolutely nuts to trade him for anything short of a franchise level player. Randolph might be on the table but I don't want him. I've always felt that he was very overrated. Oh and 16M over 3 years doesn't thrill me either.
The four guys you listed after aren't upgrades over Bass. Maybe Jason Thompson would be but it's probably pretty close to even. If we could trade straight up and save some money I guess I'd do it but when you move a player who has been on your team for more than a year you lose some chemistry. That might be worth paying for.
Someone that this blog has put be on to is Amare Stoudemire. I posted in a thread here about it and I think it would be a very bold move. Green+Lee+Bass for Amare/Brewer would be very tempting to me. It's a total "win now" move and I'm not sure that is what Danny wants but it would make us a very scary team.
Once Bradley comes back healthy, a mid-season trade of Bass, Lee, Sullinger, and a first-round pick to New Orleans for Jason Smith and Ryan Anderson.
There is no way I do that. Anderson is overrated and Smith is a Darko without any hype. Giving up all of our trading chips for mediocre bigs makes no sense. If we're trading for a center let's target Gortat or Cousins.