I’m not sure what you mean by “only one side”.
If you mean only in one direction, then I would tag it as something like maxweight:forward=5
I think this is what’s usually done for other directional access restrictions.
Are you sure this weight restriction is really only from one side?
Usually these restriction are because the road or a bridge etc was not constructed to allow heavier loads thus it must be signalled from both sides (a bridge does not allow a heavier load from one side than another).
So I would guess, that the maxweight restriction sign is just missing on the other side of the road (maybe demolished by a recent crash and not yet replaced?)
I would go there once more in a couple of days or weeks to see whether the sign was re-erected on the other side or ask the city council or whoever is responsible for the street signs in your region about the weight restriction on that street.
I don’t like the “:forward”/“:backward” suffix since the side will depend on the country (right-drive or left-drive). I prefer the “:left”/“:right” which is not ambiguous (and then similar to the tagging schema for cycleways/lanes).
I can check again whether this is the case. However, I think that the restriction is only valid in one direction to redirect heavy traffic. So if you want to go into that street with a heavy truck you would have to come from the other direction. I know it’s odd but in principle it could be.