From there I am manually copying

Those weren't the instructions you were given. You asked how to pass custom options (noxs) to Makefile.PL, so you were told to use

cpanm --look Date::Simple
perl Makefile.PL noxs
make
make test
make install
[download]

This is basically exactly what cpanm Date::Simple does, except it passes noxs to Makefile.PL.

That wouldn't work here -- noxs is specific to Date::Simple -- but the point is you weren't told to avoid the installer (by that post), and it shouldn't be surprising that doing so can lead to problems.

shared hosting where I do not have access to compilers

The options of cross-compiling or building the module on a system that is similar enough to your hosting environment have been discussed before.

There are also other hosting providers that give you shared hosting with a compiler.

It also may be worth it to pick up the thought of switching to a VPS again. Yes, it's work, and having a system with root access does mean responsibility, but it's also a lot of freedom - the freedom you're currently lacking, and that is causing you extra work (getting XS modules installed).

Perhaps an intermediate step would be for you to set up a virtual machine on your local system that you can use to "practice" on, and this can also be your test system that mirrors the production system later on. You can get Ubuntu Server 20.04 LTS here ("Manual server installation"), and getting it running inside a VM like VirtualBox is fairly easy.

It also may be worth it to pick up the thought of switching to a VPS again. Yes, it's work, and having a system with root access does mean responsibility, but it's also a lot of freedom

Yes - it probably is...
You have nailed the reasons why it hasn't happened yet. Work to migrate all the existing domains and tools that live on them, responsibility for having root access and being able to break things I don't understand and the learning curve needed to make it all happen.

There are similarities to the refactoring I am currently doing - changing an entire site from hard coded to implementing Template. It was a bullet to bite, a learning curve and a fair degree of work but actually, I've found that templates are easier and quicker to create as well as being easier to maintain later on. The reason for refactoring to templates was to be able to implement AB testing into the core of the site but other benefits have been found for which I shall be eternally grateful to The Monastery.

I plan to set up a Linux box at home and start using it for some of my everyday tasks - mostly as a learning resource. It's something that I have not got around to yet...

You have nailed the reasons why it hasn't happened yet. Work to migrate all the existing domains and tools that live on them, responsibility for having root access and being able to break things I don't understand and the learning curve needed to make it all happen.

That's understandable of course.

If you were to provide the information Corion asked for here, we could perhaps give some tips on how you could compile your modules on a different machine and then upload them to your hosting provider.

You could always just pay for hosting that provides what you need, rather than pay someone not to provide what you need.

Can you suggest how I can find out which flavour of *nix is running on the virtual server?

I've tried:

cat /etc/os-release
lsb_release -a
[download]

They tell me that the file and command respectively do not exist.

uname -r
10.0-962.3.2.lve1.5.39.el7.x86_64
[download]

Where else might I find more information so I can look into building a virtual machine or even a physical replica?

10.0-962.3.2.lve1.5.39.el7.x86_64

I think that's RHEL7 or one of the many OSes based on it, like CentOS*. Perhaps your hosting provider's support would be willing to answer that question more precisely?

Where else might I find more information so I can look into building a virtual machine or even a physical replica?

I searched for "redhat linux virtualbox" on YouTube and found a bunch of video tutorials. (Just make sure to download stuff from the official pages only.)

* Update: I see you just posted 11130454, which gives more hints: uname='linux buildfarm02.cloudlinux.com ...' suggests CloudLinux OS, a CentOS derivative.

"Where else might I find more information so I can look into building a virtual machine or even a physical replica?"

Ask the people that you pay for this service?

I remember talking about the possibility to compile them on similar architecture and copy the binaries manually.

But that's hearsay beyond my expertise, sorry.

Any ideas how I can install Net::Stripe given the compiler limitation?

You'd have to perform the download (e.g. cpanm --look Date::Simple), perl Makefile.PL and make steps of the installation on a sufficiently-similar machine. Then you'd copy the build directory over to the target machine and perfrom the make test and make install steps there.

I would first try to install the C compiler (from binaries) on the target machine.

I am trying to install Net::Stripe onto shared hosting where I do not have access to compilers

In case anybody comes across this thread node in future, I have found that Business::Stripe is a simpler implementation of the Stripe payment gateway. Plus, it is pure Perl so doesn't require a compiler and it is working whilst I solve that issue.

I am trying to install Net::Stripe onto shared hosting where I do not have access to compilers.

If it needs a compiler, its not gonna happen without a compiler.

You don't have to have a compiler uploaded on the host...

But...there are shared objects in the PadWalker directory:

.dll or .so are loadable objects

.dll or .so are loadable objects

So are the .xs and .c files source code to be compiled?

So are the .xs and .c files source code to be compiled?

That's correct. If you're not compiling them on the target machine, they need to be complied on a machine with at least the same architecture and the same Perl version as the target machine - optimally as similar to the target machine as possible, since there's a lot of moving parts involved - and only then could you copy the built files over to the target and it should work.