Or would the better option be:

1) Install the J51 template for Joomla 3 on the live site and develop it on a test page
2) Copy the entire live site to the development site and upgrade to Joomla 4
3) Update the J51 template from the 3 version to the 4 version

In this case, will the template parameters match up between the 2 versions?

use the template export json and then import the json in the new site.

Hello,
If I understood you right, you have an J3 live site.
You work on a J4 dev site (but not with the content of the J3 site) to customize the template for your needs.
After you have done that you want to transport the templates and it's parameters, custom.css and images to your j3 live site?
For me this only makes sense if you first upgrade your current J3 live site to J4.
The steps for me would be:

1. make a copy of J3 live site with Akeeba
   
2. install it on the dev environment
   
3. deactivate/uninstall all components, modules, plugins that are not J4 ready.
   
4. set default template to protostar
   
5. as I always do: I deactivate the current J3 template (if its a J51 template), not uninstalling. I know this throws an error, but I don't care^^
   
6. upgrade to J4
   
7. delete the overwrite folder of the J51 template, than installing the J4 template from J51. Acivate the template and set it to default template
   

For us this works fine, I have done it this way more than 30 times.
The J51 modules we normally uninstall especially the layerslideshow and the imagehover modules because they mostly do not work properly after an update to the J4 version of the modules.

If you are not changing from a J51 Joomla3 template to a J51 Joomla4 template than its easier.
Just uninstall the old template after setting protostar as default template and install the new J51 template J4 version after upgrade.

I hope this helps and I have not missunderstood your questions

Oliver

Thank you both for your replies. I found the export option on the Details tab of the template. This will work perfectly.