if you really want it to look "Invisible" it depend mostly on two things skills and what material you can access
IF you can find loose wool that looks just like this one then the previously meantioned needlefelting imho is a DEFINITE do this. It is Much easier than you think if you have never done it before (but avoid my mistake: take a single needle not one of the multiple needle thingies they usually
sell in the cheaper sets, this makes the work more intentional/controlled and it is still quite fast) It also has the benefit of making everything extra sturdy there and you can probably do it without haveing to open too much of your lining.
HOWEVER
If this is going to be the fancy respectable coat.. then, in my humble opinion, scavenging from your seam allowences is probably the best way. Be it for "reweaving the area" or a small patch.
The reweaving is the highest skill requirement but for a relatively small hole might be doable (there are multiple instructional videos on youtube on how to do this & it is very impressive)
The patch will be the most "obvious" version but if done well most people will never notice it, even though it might still be visible when "you" look.
BUT both option presuppose that the insides of your coat still truly have the same color, which is something you will have to see (some dyes hold up against sunlight better than others, and a coat is often worn in less sunny conditions.. but it really depends)
(I hope you treated the coat against the moths... if not yet: place it into the freezer for at least 48 hours if you have it and then clean well afterwards. Yes, I know that it was stored and there is no judgment here atr all only the point that the TLC will make probably more of a difference than the quality of the mend)
(I , personally, would TOTALLY visibly mend this but I understand that the location is a bit "off for that, but I would make it bigger note smaller (like felting differently sized polka dots in a dancing line over one shoulder and just keep the one of the hole one of the smallest), but then I am shameless and even mend pant crotch areas very overtly) (but also imagine a nice deeper Red thread sahiko patch there and another on one cuff and one matching on teh oposite Shoulder... hmmmmm.... or some meadow flowers at the hem like with a dandelion and then just a few of the parachutes dancing over your back over that area and further upwards.... however, that is not what you asked)
much love and luck in finding your materials and inspiration
S