A tub drain top can be replaced from above without cutting into the ceiling below or even going through the opposite wall, but whether an inexperienced DIYer can do it is questionable. Look up YouTube videos on this.
The top is unscrewed with a tool that is inserted into the drain and locks onto the cross hatch. Under the tub is a gasket between the bottom of the tub and a flange on the drain below. This is replaced with a new one of the right size. Plumber's putty is used to seal between the underside of new top and the tub. The new top screws on and is tightened with the same tool used to remove the old top.
There are possibly different sized tops and possibly different threads, so measure the diameter and take a pic to the plumbing supply. You might have to remove the old one and take it with you.
If the drain top has no cross hatch, there surely must be a tool which when inserted into the drain, clamps to the inside and allows unscrewing the drain top. If there is no special tool for this, one must try to improvise, e.g., https://youtu.be/0poO0fky7Mw . Having seen this video, I'd try my Vaughn Bearclaw right angle prybar, but maybe a hammer would be better.
In the video above, she didn't change the gasket so maybe that can't be done that easily. I have seen a video in which that was changed from the tub side. The gasket is not to seal water but to prevent damage to the underside of the tub and to distribute the clamping force so as to squeeze the plumber's putty uniformly. The plumber's putty is the seal.
There is a tool called an internal pipe wrench.
If it were me, I'd try to get a replacement with a cross hatch. The wrench for that is widely available.
https://youtu.be/8neK7Nj0Xs8 shows multiple tools. This guy has hilariously failed to understand how people keep "wrongly" saying removal of a cross hatch drain top can be done with pliers! He imagines using it handles down!?.