I was using a Seagate 2TB portable HDD for almost a year without issues.
Didnt have a particular name,was just the cheapest I could buy.
But I only used it to store roms,and had it formatted to NTFS,I booted Batocera from a flash drive.
Having said that I always had back ups of roms incase it failed.
Now Im not recommending you get the cheapest 2TB Seagate or WD portable drive you can get,Im just sharing my experience.
Are you using it for roms only or the whole Batocera OS as well as roms?
Since you are willing to pay a little more perhaps look at those WD Black game drives. Theres good reviews on them.
I use a WD Black 2TB Game Drive for my PS5,to hold PS4 games. Had that for a while,no issues as yet.
As for SSD’s,they ofcourse will be more expensive,not sure you would be able to get a legit brand 2TB for the money you want to spend.
Even then from what I read using an SSD will only give slightly better boot time (if you are booting Batocera from it) and slightly faster loading of disc based games.Performace wise for emulation it wont be better for most systems..perhaps only for PS3 but not sure.
However recently Ive switched to just using the internal HDD of the PC. Batocera and roms are now all running off that,just cause I wanted it as a dedicated system only.
Perhaps others have better recommendations?