You can leave the mushroom intact, by the way. In the picture, you can see what I mean.
Actually you only need the porous part, and it is possible to cut the band-aid quite thin. The pictures were obviously taken for demonstration purposes. The injury was already healed and well at that point.
Now please keep in mind that this is an emergency remedy. If you have a first aid kit, please, at least use proper disinfectant before treating any wound at all. I am not to be made liable for any healing impairment or deterioration of the healing process to to the application of natural remedies. I share this as a merely cultural evaluation and for entertainment purposes exclusively.
That said, the use of birch polypore in indigenous cultural heritage as a natural remedy goes back to the paleolithic age. A later, and rather famous find from the late neolithic/ copper age, showed the multifunctional use of birch polypore. Otzi, the "iceman" carried, amongst other things, a piece of birch polypore as tinder, but probably also as medicinal. Studies have found traces of the mushroom in his digestive tract:
https://www.sciencedirect.com/science/article/abs/pii/S1879981718300883,
https://www.sciencedirect.com/science/article/abs/pii/S0378874114003183
and presumeably he had taken this remedy for its anti-inflammatory and immune system modulating properties:
https://pmc.ncbi.nlm.nih.gov/articles/PMC5380686/
The mushroom has many uses and preparation methods. The dried powder was also used as a painkiller (see: https://pmc.ncbi.nlm.nih.gov/articles/PMC5380686/).
Relevant for the use as a band-aid are of course said anti-inflammatory properties. Responsible for these properties are mainly the Tri-Terpenes and Tri-Terpenoids. On the other hand, the wound-healing process might also be furthered by the poly-saccharids in its composition. Hydrogels with Saccharides and Glukose are a therapeutic remedy for necrosis even today. At least, those polysaccharides in the mushroom are said to offer a moisturizing environment for the wound to heal. For the same reasons, the fresh tissue from Fomitopsis Betulina can be used for skin care and hygiene. The dried mushroom must be rehydrated first. The dried powder might be best for the purpose.
So, while obviously cutting a band-aid from some old shroom in the woods can, in that situation, only be an emergency remedy, with a bit more attention to care, hygiene and attention, it could be far more than just that. The mushroom shows a lot of promising characteristics in a lot of fields of use.
Please harvest the mushroom sustainably. Take only what you need, and if at all possible, keep the mycelium intact. As a rule of thumb, take just 20% of the mushroom in order for it to regrow. Leave the Trama intact, too. Of course there is nothing wrong with harvesting a whole fruiting body where the mushroom is abundant, too. Just use common sense is all.
All the best, and take good care!