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It’s good you already provided very helpful information – an internal storage nearly full is a prime suspect for why a smartphone turns laggy. So you already mentioned what I would recommend:
I would recommend you don’t save single-digit Euros and go for at least a 128 or 256 GB card and then make sure you either leave it formatted as it is or format it as 
portable storage (and not 
phone storage = internal storage), see ✏ A little guide to... using SD cards. Some random experience with existing cards can be found here ✏ List of compatible micro SD / SDXC cards for FP3
Of course, you could also move some dozen GBs to a different storage medium (computer, cloud, hard drive etc.), but I would recommend to do that in addition to a card, not as an alternative – this way, you can still access your photos on your FP3. 
A factory reset can help to get rid of hard-to-find problems that might occupy system ressources, but you can still go to that option after you have gotten yourself an SD card and moved dozens of GB from the internal storage to the card.
A battery replacement is probably not helping with making the device fast again, but once you are at it, you could check yours for signs of bloating: Take it out of the device, place the battery flat on a really flat surface and see if you can make it spin. If it’s easy to make it spin, it is probably showing first signs of bloating and should be replaced – but only for security reasons.