Provider has to take it down. The vendor making the shirt can simply file a dmca counterclaim (basically stating that the dmca claim is invalid)
Then the provider puts the content back up and the courts decide.
I don't believe that's true. I think they must make the material available after a proper counter-notification, or they lose their liability shield as per the exception here. IANAL, just reading the text of the law at face value.
(g) REPLACEMENT OF REMOVED OR DISABLED MATERIAL AND
LIMITATION ON OTHER LIABILITY.—
(1) NO LIABILITY FOR TAKING DOWN GENERALLY.—Subject
to paragraph (2), a service provider shall not be liable to any
person for any claim based on the service provider’s good faith
disabling of access to, or removal of, material or activity claimed
to be infringing or based on facts or circumstances from which
infringing activity is apparent, regardless of whether the material or activity is ultimately determined to be infringing.
(2) EXCEPTION.—Paragraph (1) shall not apply with
respect to material residing at the direction of a subscriber
of the service provider on a system or network controlled or
operated by or for the service provider that is removed, or
to which access is disabled by the service provider, pursuant
to a notice provided under subsection (c)(1)(C), unless the service provider—
(A) takes reasonable steps promptly to notify the
subscriber that it has removed or disabled access to the
material;
Applicability.
PUBLIC LAW 105–304—OCT. 28, 1998 112 STAT. 2883
(B) upon receipt of a counter notification described
in paragraph (3), promptly provides the person who provided the notification under subsection (c)(1)(C) with a
copy of the counter notification, and informs that person
that it will replace the removed material or cease disabling
access to it in 10 business days; and
(C) replaces the removed material and ceases disabling access to it not less than 10, nor more than 14,
business days following receipt of the counter notice, unless
its designated agent first receives notice from the person
who submitted the notification under subsection (c)(1)(C)
that such person has filed an action seeking a court order
to restrain the subscriber from engaging in infringing activity relating to the material on the service provider’s system
or network.
They lose their liability shield for the act of taking the content down, but that liability shield is of limited value to begin with because they'd typically be well within their rights to take any content down for any reason (or no reason).