Could the Cubs sign the free-agent third baseman?

Last offseason, the Cubs were reportedly “in” on free agent third baseman Alex Bregman. They got outbid by the Red Sox.
A couple of weeks ago, I posted this article regarding Bregman’s free agency this offseason after he opted out of his deal in Boston.
Now, per Patrick Mooney and Sahadev Sharma in The Athletic, the Cubs once again have interest in Bregman:
After a failed bid to sign Bregman last spring training, the Cubs have shown renewed interest in the All-Star third baseman, according to sources briefed on the club’s offseason planning.
While certain elements of the roster have changed — and the organization has different momentum from its first playoff appearance since 2020 — the logic that drove the Cubs to pursue Bregman then still applies now.
The article goes on to describe the contract offer prepared for Bregman a year ago by Cubs President of Baseball Operations Jed Hoyer:
In making his presentation to the Ricketts family, Cubs president of baseball operations Jed Hoyer identified Bregman as a great fit with this core group and a solid long-term investment. Chicago’s ownership group authorized Hoyer to make Bregman a four-year, $115 million offer that included opt-out clauses after the second and third seasons, according to sources briefed on the negotiations.
That was, at least, in the proverbial “ballpark.” The deal Bregman eventually signed with the Red Sox was for three years, $120 million, with two opt-outs, one of which he exercised. So the deal was for slightly more total money, but a higher AAV.
Bregman had a productive season in Boston, batting .273/.360/.462 with 18 home runs in 114 games. You’ll notice that means he missed 48 games — he was out about seven weeks with a quad injury, and though his numbers did decline somewhat on his return (.938 OPS and 11 HR in 51 games before the injury, .724 OPS and 7 HR in 63 games after), overall it was a 3.5 bWAR season. He’s still a good defender at third base and won a Gold Glove in 2024.
So would you do this? Signing Bregman, who will turn 32 right around Opening Day, probably would require at least a four-year deal and probably five. The article I posted last month noted that MLB Trade Rumors projects Bregman for six years and $160 million. I don’t think I’d commit six years; perhaps five and a vesting option?
If the Cubs signed Bregman, that would leave Matt Shaw open for trade. Shaw could be an attractive piece in a deal for a starting pitcher.
I think I’d do those things. Would you?