There does not seem to exist a designated feature for this as on Steam. You could potentially try to add the web version into the install designation where the app usually puts downloaded games into (or create an alternate install location for where your web game is), but I would not recommend this, if it even works at all. The app is meant to add and display games from itch.io that have been made compatible with it, so just adding outside sources may not result in anything working well.

That being said, you can try to check the install location after you downloaded the specific game again and try to add in your save files from the web version, at least.

Edit: I will not make a new post for this as it is not specifically tied to this topic matter anymore, but to anyone reading this: I made a mistake here and the statement 'needs to be made compatible with the itch app' is not true. You can upload it specifically with the butler tool to have smaller updates for both you and your customers, but you can install items which have been uploaded directly to the web page as well.

Is there even a feature to verify an installation? If the app manages the installation I would expect such a feature. And from what I read about the app, there is not really a thing like making a game compatible with the app. Not as a requirement that is. I believe one could create a manifest for the app, but that is not mandatory for it to work.