If the recharge system functions then yes, the cell can react badly. Most likely by swelling and leaking. There is an argument that a very slow trickle charge to an
alkaline will result in recharging of the cell but the physical structure and internal chemical make up were not designed for recharging. Lithium non-rechargeable batteries do not like recharging and can result in a runway chemical reaction destroying the Battery and probably the watch module. Lithium batteries HATE two things:
1). HEAT. (So charging a non rechargeable lithium is lethal to the Battery.
2) deep drain with no recharge (I.e. use until dead then recharge attempt)
My Star Wars EPD recharge circuit failed at the 3 year mark. The watch went to Seiko japan who replaced the charging circuit with a newer mark ‘2’ version 3 years ago and to my surprise, for free, except shipping. Holds a perfect full charge always now. Of course I leave it near the window when not in use, not at the window and never let it cycle to dead.
I would hold off, find a rechargeable 1616 lithium battery OR cut the recharge circuit wire for a permanent solution and only buy non rechargeable batteries. Careful as the recharge circuit may be involved with current transfer to the watch.
Amazon Canada (assume USA also)
https://www.amazon.ca/Casio-CTL1616-Genuine-Rechargeable-Battery/dp/B0080GQBTUGeoff