Battery gets old whenever you are using it or not. You can leave a fully charged battery on the table for few years and it's capacity will drop anyway (not necessarily so much like when keeping it in the laptop, which means leaving it to self-discharge to 95% and recharge to 100% and over again, but still), it's unstoppable chemical process. Battery count is not what is most important. Important is battery health and that's what matters most.
I work with RC models and I have "some" experience with Li-Ion/Pol cells and I can tell you it's better to use the battery than not using it. Pure theoretically, if you want to keep the battery as most fresh as possible, you can charge the battery to 50%, take it out of the laptop and keep it in the fridge. This is most efficient way to slow down battery aging.
Laptops have batteries because they are portable devices and batteries are intended to be used. Keeping the battery in the laptop fully charged does not save the battery at all, it's even worse than using it (using the laptop on battery).
As mentioned, if you keep the battery charged and never use the laptop on battery (discharge it), you may notice it's charge slowly drops from 100% to cca 95% even when the laptop is powered by adapter. that's the charger taking care of the battery by stopping charging full battery.
as a bottomline and already a bit off-topic I (yeah I know my reputation here is not high, I just joined) recommend to use your laptop as a portable device - use the battery, don't go down to 0% charge (and if you have to, don't leave the battery discharged for days - it will say BYE and travel to **** wifh a puffy face), don't keep it constantly on AC. just use it as designed.
back to the topic - you are saying you don't use the laptop on battery, therefore there will be absolutely no effect when you swap HDD for SSD. battery is not used at all while on AC