does Bob have a claim against Alice for not enforcing his exclusivity? Does he have a claim against Carol?
The licensee primarily has a claim against the infringer of the patent, but see Ind. Wireless Co. v. Radio Corp., 269 U.S. 459, 468 (1926):
The presence of the owner of the patent as a party is indispensable
not only to give jurisdiction under the patent laws but also, in most
cases, to enable the alleged infringer to respond in one action to all
claims of infringement for his act [...]
which is why
[i]f the owner of a patent, being within the jurisdiction, refuses or
is unable to join an exclusive licensee as co-plaintiff, the licensee
may make him a party defendant by process and he will be lined up by
the court in the party character which he should assume.
Id at 469.
That "party character" would be co-plaintiff. See Gentesix v. Board of Regents of Univ. of Texas Sys., 966 F.3d 1316, 1323 (2020):
[A] patentee who refuses to voluntarily join an infringement action
initiated by its exclusive licensee can ordinarily be joined as an
involuntary plaintiff under Rule 19(a).
For Bob to have a claim against the owner of the patent, the terms of the license agreement would need to establish the owner's responsibility. For instance, Ind. at 469 notes that
exclusive licenses frequently contain express covenants by the
patent-owner and licensor to sue infringers that expressly cast upon
the former the affirmative duty of initiating and bearing the expense
of the litigation.
This reflects that the owner a priori has no duty toward the licensee in regard to enforcement of exclusivity.