OK on the opacity.
I believe there are ways to cache the Open Street Map (OSM) tiles, but we haven't implemented any in MapRun, as the main-stream usage is to display an "Orienteering Map" as an overlay on the underlying OSM/Satellite base. To ensure you can go off-line, we hold the overlay locally as a KMZ file on the phone.
If you actually want to display the background of your course as OSM, and go off-line with that, the current approach in MapRun would be to create a KMZ file from the OSM imagery and use that as the "overlay".
There are various tools for producing KMZ's ... but a simple approach that might be "good enough" would be to get an image of the area in OSM (even with a simple screen capture on a PC/Mac), and then convert that image into a KMZ using function at the MapRun console:
https://console.maprun.net/#/kmz_createPeter