Disclaimer: I am not a lawyer, much less your lawyer, and this answer does not provide competent legal advice.
The software provider claims that the process to setup/install/configure/implement the application is proprietary.
That's plausible, but irrelevant. The software provider has executed their proprietary procedure on your server. It's now done. That the process is proprietary does not in itself place any obligations on you.
They consider the implementation process as copyrighted original creative work.
That could conceivably be supportable (though more likely not), in the same sense that a performance of a symphony can be copyrighted. But that's also irrelevant. Any such copyright they have protects against you copying the performance, which you have no way to do anyway, having not recorded it. Copying the contents of the server after the fact is not the same thing as copying the installation process itself.
They've also stated that their installation includes proprietary copyrighted components belonging to them.
That could be true.
They won't allow me to use a third party to backup the server as it will breach the copyright of their work/install.
Meaning, you've clarified, that they demand that you not engage anyone else for backup services, not that they can physically or electronically prevent you from doing so.
In the first place, then, you should consider whether they have a leg to stand on under copyright law generally. You didn't say where you are, but in the U.S., the Copyright Act permits you to make a backup ("archival") copy of computer software, though not necessarily of other digital materials.
This is where you review your agreement with the software provider, or have your lawyer do so. It is possible that the agreement holds that you waive your rights to make backups, or that it gives the software provider an exclusive right to provide backup services. If it does not do either of these things, and if U.S. law or a law with similar provision for software backups applies, then no GPL-specific analysis is required.
But if you're not so lucky,
Does the GPL-3.0 license allow them to place this restriction?
Does the GPL permit them to forbid you to make copies of their proprietary, non-GPL components? The GPL does not apply to such components.
Does the GPL permit them to create a proprietary derivative of GPL software? Yes, it does, but it does not permit them to convey a derivative of GPL software to a third party (you) under terms other than the GPL, which permits you to make copies as you like. Furthermore, the provider would be obligated by their license to provide source to you on request.
Does the GPL prevent them from engineering a situation in which you have a copy of a proprietary GPL derivative whose rights they own? No, it does not, though some gymnastics would be required. This would involve provisions along these lines in your agreement with the provider: you agree that the provider is creating a derivative of the GPL software for you as a work-for-hire, and you agree that after they install this derivative on your server, you assign all your rights in the result to them.
Much depends on the details of your agreement with the software provider. In addition to any provisions specifically about backups and copyright, you should pay attention to exactly what the provider agreed to install. If they have installed more or different components than you specified, then they may have violated / failed to fulfill their end of the agreement. Do read very carefully, however, as there could be a huge difference here between, say, "PostgreSQL Server 15" and "Joe's Software Shack database server bundle featuring PostgreSQL 15".
Among the options that may or may not be available to you are:
Just go ahead with engaging whomever you want to make backups. This is advisable only if it is permissible under applicable copyright law and in light of the details of your agreement the the provider (not the GPL).
Ask the provider to identify their proprietary components, so that you can remove them. This carries a risk of rendering the software you wanted inoperable, which might then roll over to ...
Demand that the provider reinstall with (only) the software the agreement specifies. Of course, this assumes that the provider has done differently in the first place, which might not be the case.
Avoid performing backups (not recommended), or limit backups to pieces that you can be sure do not belong to the provider. The database, for example.
Engage the software provider to perform backups.
Wipe the server, and engage a more reputable software provider for the next one. Note that you probably still need to pay the original software provider in this case.