The simplest approach seems to be to just increase damage at some BAB points.
Not as fast as doubling every 5; each attack is 25% less likely to hit than the previous, generally, so doubling at BAB 6, tripling at BAB 11, and quadrupling at BAB 16 would be too much.
You could simply give free Vital Strike feats at the appropriate BAB, but this is also less damage increase than more attacks are worth, largely due to Vital Strike only scaling the dice and not the static values.
Also consider how this may interact with things like Rapid Shot - you would probably want to remove this.
And damage reduction, which matters less when the individual attacks are bigger. And the flexibility to use combat maneuvers as part of a full attack sequence. This also decreases the value of combat maneuvers at higher level, since what you are giving up is much greater. (Unless you implement something where you give up only part of the damage to perform the combat maneuver, which is convoluted.)
An alternate form would be to have hp and damage generally not scale significantly as characters level up, but that would require rebuilding a lot.