I'm going to ramble for a minute
..
It sounds like you have a pretty good understanding. I wrote a blog post (from 2013, but the differences are still the same) that talks about the differences between Responsive Design and ShopSite's Mobile CGI.
http://laurenhillsdesign.blogspot.com/2013/02/responsive-design-vs-mobile-cgis.html
Mainly, the Mobile CGI is a great solution to
quickly turn your website into a mobile friendly website. However, long term, Responsive Design is typically the way to go. You could use the mobile CGI, and start working on creating (or choosing) a responsive design template, and release that template when you are ready. The reason Responsive Design is typically better in the long run is exactly what you mentioned "over the years I've made a good number of edits". With Responsive Design, making those edits will apply to all versions of your website, desktop, tablet and mobile. With the Mobile CGI, you have 2 sets of templates, so if you make custom settings or use custom fields in your template, you will need to make those edits in both sets of templates.
ShopSite does have some Responsive Design templates that are built into the software, and many more that are being added in the about-to-be-released version. Of the soon-to-be-released templates, there is one called "Trace Colored" that has a very similar layout to what you have now, so it would be a pretty good option to switch to if you wanted to switch your website to use a built in theme that is already Responsive. It would take quite a bit of work to turn your existing template into a responsive template because responsive templates typically have at least 3 layouts built into 1, so at least 2 more layouts would need to be built into/added into your templates.
With the "fixed size images", does this mean that they are actually a different, larger size, and you use code to make the image smaller ( see you do this with the color swatches in your product descriptions, but those swatches are all the same size and small, so there would be an easy work around for those)? If this is the case, then is the image on its own line or is it left or right aligned with text wrapping around? If your images are mostly all on their own lines (like how your products are laid out, where you have the product image on its own line, then the product name description and price are displayed below), then you can do responsive design. It looks like the images have a more complex layout on your home page (as most sites do), so those images and their layout would likely need to be setup for responsive design specific for that home page (and any other pages that have an image layout similar to your home page).