Memory is a complete waste of money for a computer this old. If memory was a problem, then you would have upgraded the memory years ago. Plus Macs are very picky about the memory they use so definitely do not buy RAM from Amazon since there are a lot of fake & scam products on Amazon (you cannot trust the ratings or reviews either). @John Galt has been providing you with very good advice.
The biggest performance boot can be obtained by upgrading the internal hard drive to an SSD. The best choice for an SSD is the Crucial MX500 series SSD as it is the best compromise on price & performance, but avoid the Crucial BX500 series since they are absolute junk and may even be slower than your current hard drive.
Plus if you do decide to upgrade to an SSD, then you may also need to replace the internal hard drive SATA Cable if you encounter problems. I would avoid the Samsung SSDs, especially the EVO series since I have seen a lot of people report compatibility issues on some Macs, plus the Samsung SSDs are usually much more expensive. The Crucial MX500 series SSD I mentioned is probably the best option.
Also, why get a 2TB SSD when you have been happily using a 750GB size drive? Are you really going to be storing that much extra data on it?
Even macOS 10.15.x is really old now. Most third party app developers will only support the last three versions of macOS these days (currently Sonoma, Ventura, Monterey). This includes most of the popular web browsers. It won't be long before the third party browsers will no longer be able to access the Internet.
Depending on how you want to utilize this laptop, it is possible to install Linux Mint on it instead which would give you an up to date, non-Apple OS capable of using most of the popular web browsers. You won't be able to run any of the other popular third party software so you would need to find a third party alternative to work for any other things you want to do with this laptop. There is a lot of free & open source software available in the Linux Mint software repositories. Linux works well with just 4GB of RAM installed as long as you are not doing anything too memory intensive. You can easily have dozens of windows & tabs open in a web browser.
Linux does require you to learn a new OS, but it is a great way to extend the useful life of older computers. However, Linux is not for everyone. You can take Linux Mint for a test drive by selecting "Live" mode from the Linux Mint installer. Linux Mint will run extremely slow this way, but at least you will be able to see what it is like with the default installed apps.
Just make sure to create a bootable macOS 10.15.7 USB installer before doing anything else so you can easily install macOS if you change your mind about Linux Mint.
I also like to keep older computers running for as long as possible. In fact I have several of them from about 2006/2007 with the old Intel Core 2 Duo CPUs which are able to allow me to do basics tasks with just 4GB or RAM.....mostly surfing the Internet and I tend to keep a lot of tabs/windows open. I would never invest any more money into them other than an SSD which I could later remove and use for external storage. Memory is a complete waste since memory will only work in that particular computer. Only you can decide what to do.
FYI, newer Retina model Macs do not run very well without a battery connected. It is only the older non-Retina models that could run just fine without a battery connected...although it may trigger the fans to run at high speed since the battery temperature sensor would not be communicating.