Alaska and New Mexico are both members of the Driver's License Compact so the information regarding your failure to submit for the breath analyzer (BAC) test should be reported back to your home state of NM. Along with the Alaska violation information, the AK courts or DMV will probably also submit to the New Mexico Motor Vehicle Division what they are requiring of you, including the SR-22.

If New Mexico does not require SR-22s for various infractions then you may need to discuss with Alaska what other NM insurance they may allow instead of the SR-22 in order to comply with their requirements on you.

As for your New Mexico driver's license renewal, you will need to check with the NM MVD to find out if you can renew your license without complying with the penalties placed on you by Alaska, such as the SR-22. It might be that if your license is currently suspended in AK, that NM will require you to sort out those issues before renewing your license.