"cautiously agree" is correct and usable in written English.
You can use it when you don't want to duplicate an offer or agreement completely but you want to accept it with reservations. For example, "I cautiously agree to the terms of the contract."
While no one is doing cartwheels, analysts cautiously agree with this sunnier picture.
If we are to cautiously agree that women are more at ease with discussing emotions, and therefore more comfortable with the idea of embracing their emotional response to music, then it is logical to assume that the songs which aim for the emotional jugular might appeal more to women than to men.
Other researchers cautiously agree, but stress that more meaningful recoveries may demand a combination of approaches.
And those who work with prisoners (and ex-cons) tend to at least cautiously agree.
The Olivias and Walters cautiously agree that they need to put aside their differences to save both worlds, apparently unaware of Peter's disappearance.
Mr. Hage cautiously agreed.
The hostess cautiously agreed, saying, "It's hard as a mom to get off the grid".
The teenager cautiously agreed that "Magic" sounded OK - "I just didn't think it would stick," he later said.
He cautiously agreed to an anonymous interview – he's worried that being openly associated with Project Prevention could cause problems for his family.
But Lawrence and Evans prevail, and Claudette cautiously agrees to become the focus of a march movement to desegregate the downtown business district.
The manuscript was rejected by the original publisher; another house cautiously agreed to take it on, offering a $1,500 advance to be shared with the book's French co-authors, Louisette Bertholle and Simone Beck.
