The bridge 'incidents' were extremely isolated - i wouldn't worry too much.
No reason you can't run the B and E thru body - just make sure there's enough untapered string length to do it properly - at least 36" from ball-end to taper should be fine to get it past the nut (just make sure there's still enough to get as far as the tuners!

)
I've had .136 B strings thru the bridge on my Tribby, so you should be just fine with any strings the same thickness or thinner.
Report back plz - let us know how you got on....
James.