First, I must admit that I was pleasantly surprised by the Dunleavy signing. I doubted that the Bulls would add anyone above the vet minimum in free agency, and Dunleavy (health permitting) is one of the best backups in the league. He's a great value signing given that his most comparable players (e.g. Korver, Budinger, and Kevin Martin) have each been getting more years and roughly double the annual salary, and his presence should ensure that Deng doesn't lead the league in minutes per game for the third consecutive year.
Despite this encouraging move, there are still 3 issues that I see with the current roster:
1. No reliable backup center to lessen the load on Noah
2. No low-post scorer
3. No secondary shot creator
I believe that a team needs at least one of those final two elements (low-post scorer & secondary shot creator) in order to sustain an effective offense during the playoffs, particularly in late-game situations.
While all 3 of the identified issues could be rectified by a trade for Aldridge or Love without giving up Noah, I'd rather focus on moves that I'm nearly certain the team could make rather than speculating on what it would take to pull off a blockbuster trade. I'm also mindful of the fact that the Bulls haven't had a consistent low-post scorer in a long time (since Curry if I'm remembering correctly), and those guys tend to be quite costly to obtain, whereas the Bulls had a player who could fill the secondary shot creator role just last year in Nate Robinson. With that said, here's what I would do:
A. Trade Teague + TPE for Ayon + Dunleavy. Contact the Bucks and arrange to trade Teague and part of the Korver TPE for Ayon and Dunleavy, rather than spending the MMLE on Dunleavy. The Bucks should gladly accept this offer, as 3 years of Teague has more value to them than 1 year of Ayon, who is 28 and presently their 3rd-string center (after Sanders and Udoh, not to mention Henson once he adds enough strength to play his ultimate position of C as opposed to PF). Ayon is a solid rebounder and defender who has no range offensively but is a surprisingly good passer and crafty enough to get some easy buckets around the rim. He's a significant upgrade over Nazr.
If you're familiar with the new CBA, you know that teams over the luxury tax apron can't complete sign-and-trades, and therefore the Bulls cannot utilize this method of acquiring Dunleavy as the roster now stands. This leads us to part B:
B. Amnesty Boozer. It's simply the move that makes the most sense, competitively and financially. Boozer remains a good rebounder, but he's terrible defensively and his offense has slipped to the point that he's not better than Taj on that end. The Bulls scored 102 points per 100 possessions with Boozer (worst among the team's top 8 players) versus 107 pts/100 with Taj (best among rotation players). When Boozer's quirky jumper isn't on, he's not worth having on the floor.
The payroll currently projects to about $80M, or $8M+ over the tax line. That's approximately $13M in luxury tax payments that the Bulls stand to make at the end of the season, in addition to lacking access to the MLE and sign-and-trades. If the Bulls amnesty Boozer, they can turn the Dunleavy acquisition into a sign-and-trade AND have the MLE left to spend while remaining under the tax. Let's say that Boozer gets $5M/yr from some team and the Bulls apply that $5M to the roster by using the MLE. Then the Bulls will still owe $80M to players, but the payroll is officially $70M, so they save the $13M in luxury tax payments and receive the tax distribution to non-payers. Amnestying Boozer isn't the equivalent of paying Boozer to play for another team; it's effectively saving the full amount of Boozer's contract.
C. Sign Nate for 2 guaranteed years and a total of $5M-$6M. No team is prioritizing Nate, and it sounds like he'd gladly accept a multi-year deal from the Bulls at the MMLE amount. The Bulls could probably even add a 3rd unguaranteed year to get some extra upside out of the deal. Last year, the Kirk + Nate backcourt was phenomenal, often against the other team's starters. That lineup combination was +15 pts/100 in the regular season and a whopping +36 pts/100 in 46 playoff minutes. The 300 minutes they played together last year isn't a huge sample, but it suggests that they'd give the Bulls a major edge against most teams coming off the bench.
D. Add a backup PF to play 10-15 minutes per game. The Bulls would have $2M-$2.5M of the MLE left to spend, or they could sign a guy for the minimum and keep more financial wiggle room under the tax in case a trade presents itself later on. I'd offer the remaining amount to Brandan Wright considering his low-post scoring ability, but assuming that he could get more money and a larger role elsewhere, I'd settle for 1 year of Elton Brand or bringing in Josh Harrellson at the minimum (with the first year guaranteed and hopefully an unguaranteed year or two thereafter). I think Brand and Harrellson would be good fits because they defend and rebound, unlike Vlad Radmanovic, who was a decent signing in a vacuum but is the type of player that Thibs can't trust.
This would be the resulting primary rotation:
Rose / Hinrich
Butler / Nate
Deng / Dunleavy
Taj / [Brand or Harrellson] / Deng
Noah / Ayon
With regard to the bench, in addition to Hinrich and Nate's success together, the Dunleavy / Ayon combination was also +36 pts/100 in 113 minutes in Milwaukee. I'd think that the starting lineup would suffocate opponents with defense, and then the bench would have a tendency to blow the game open in the second quarter, especially considering how many weak teams there should be this year. Then Snell, Murphy, and Nazr can get some time on the court in blowouts to develop in the case of the rookies and to be prepared in case of injury.
The Bulls would also be more versatile than in previous years, as losing Boozer would free them up to go small more often with Deng sliding over to PF. The Bulls could more effectively play offense/defense using substitutions, replacing Taj with Nate or Dunleavy when more offensive firepower is desirable, and everyone other than Nate can competently defend multiple positions.