I think there is a big difference between the options available at Muji and a proper suit. The latter will probably serve you much better in the long run for around the same price.So I'd check out the suit shop options first, as it is only a few thousand yen more, and they may be able to tailor it in time. Even if they can't tailor in time, neither can Muji, so you can always keep that as a backup.
Also, your gf's grandfather's funeral is a time to be as formal as possible. All the men will be in proper black suits and you wouldn't want any reason to stand out more than you already do just by attending.
BTW, I'd confirm that you're even going first. As you metioned, check with the family, and make sure its ok with them and not just your gf. Also, are you aware of the procedures for a Japanese funeral. The family sits up front seperated by gender, so likely you will be sitting alone with the regular attendees and will not be with your gf during the service. Make sure you are briefed on what to expect as it is very different from a western funeral, and could be quite an uncomfortable experience to go in unprepared.
See these pages for more info:
http://www.japan-guide.com/e/e2060.html
http://tanutech.com/japan/jfunerals.html