just add another AP to the mix and hardwire it??
Yes.. but to gain the advantage of roaming / handoff.. use two EAP245 ..
Would the EAP245/Airport Extreme handle roaming ok?
No.. Roaming uses a whole new set of wireless standards.. 802.11k, v, r as well as a heck of a lot of tweaking. Mixing brands is definitely NOT a good idea. Extreme is 8-9 year old design assuming we are talking the AC model.. and Apple have not updated it since release other than security and bug fixes. Even though the phones etc moved to roaming standards. Hence writing was on the wall for the Apple Airports long before they chopped off the production line.
What I suggest is keeping the Extreme as your router.. that is fine. Turn off the wireless.
Plug one EAP245 into the Extreme upstairs and plug another one into ethernet downstairs.. You can add a small gigabit switch there if you don't have another outlet. They are POE so you could get a POE switch or just use the supplied POE adapter.
TP-Link line of semi-pro products is not well known.. they are obviously trying to compete with Ubiquiti which is also a good solution.. albeit a more complicated product without a built in GUI. In a business situation I would go ubiquiti. They are better at updating products.
The TP-Link are just as good wifi and design wise.. but without the level of support. Hugely easier to configure though.
Here is a good review.. one of the few decent tests I found.
https://dongknows.com/tp-link-eap245-v3-omada-poe-access-point-review/
I do have a set of EAP225 which are slightly cheaper 2x2 version but also very good.
The controller was so cheap I also bought one.. IMHO it works really well.
I have also used Ubiquiti.. but for home use I don't think it is as easy.. though you have more IT skills than most so be worth a look.