Out of curiosity, what's your concept to integrate the two? I think it would be far easier to add a BT simulation to SR...not sure about going the other way (though my understanding of non-BT tech within the setting is a little limited).
AJC
1st thing that came to mind is expensive but, get some sustaining foci with illusion spells on them and people actually become battlemechs. Like those life-size chess sets from
history of the world. Find a park and have fun. It could be done far cheaper with augmented reality and you could literally use the entire city as a map.
2nd thing I was thinking which would be probably even more expensive and might not be possible but build life-size replicas of the Mechs that actually work and use them that way.
3rd thing would be using magical or technological holograms in your own place to make it look like a command center for a Mech and connect with other people through something like Skype.
Things could get quite interesting if somebody has the technical computer cyber where in their head. :-)
As for the other way around in battletech I think some of the worlds, more advanced worlds, have augmented reality, I think. You could do augmented reality shadowrun where you can look like an actual troll and sling spells to augmented reality. Role-playing games might be quite popular because of the desire to check out of the real world because conditions in battletech.