I would contact your SENA representative and ask him or her what they suggest. Most of these rights' organizations have reciprocal deals with their various international counterparts. It looks like SENA has a deal with PPL in the UK (which in-turn has a reciprocal deal with the AFM, AFTRA, and the AARC). They do not look like they are directly connected with either ASCAP or BMI, but they may be through their relationship with the AFM. Before redundantly convoluting things and running off to join another rights organization, I would peruse your SENA materials, website, or ask someone on-staff who else they may have a relationship with in the US and Canada...
GJ