Is there a word that a best man could use to refer to his relationship with the groom, or better yet the bridegroom? No, not really.
The more precise and complete term for groom is bridegroom, the former does not refer exclusively to the bride's soon-to-be-husband as it can also mean the person who tends to the horses and stables. If outside the church and before the wedding reception the best man referred to his friend as “my groom” people might think he were talking about his employee; or, they might infer that the OP is speaking about his male partner, and consequently a same-sex marriage was forthcoming. Might... if they didn't know anything about the OP.
Wikipedia explains:
A bridegroom (sometimes shortened to groom) is a man who will soon or has recently been married. A bridegroom is typically attended by a best man and groomsmen.
Etymology
The first mention of the term bridegroom dates to 1604, from the Old English brȳdguma, a compound of brȳd (bride) and guma (man, human being, hero). It is related to the Old Saxon brūdigomo, the Old High German brūtigomo, the German Bräutigam, and the Old Norse brúðgumi
Interestingly, the term best man, first noted in 1814, is Scottish in origin and replaced the older groomsman used since the 1690s.
Traditionally, the groom chose a man who would make sure the bride's family would not recapture her during the wedding ceremony, he had to stand beside the groom, alert and well-armed. In addition, he was expected to continue his duty outside the newly-weds' home (presumably until the couple had consummated their marriage). The "best" part of that title refers to his skill with a sword, should the need arise.
As can be seen, the term bridegroom is a compound word meaning the man or hero of the bride. It therefore follows that the groom-to-be cannot also be the ‘man’ or ‘hero’ of the best man. Today the best man is left with the following choices:
- I am best man to [First and last name]
- I am his best man (or traditionally, groomsman)
- I am the bridgroom's best man