POST TESTING EDIT: All of this was theoretical. After putting it all together, it severely under-performs. My frostbite is under-leveled, but the rest of the build is there. Part of the problem is that you go through the magazine super fast. It has been effecting my ability to capitalize on crit procs. Otherwise, the damage is just not at the level I would consider acceptable for a late game weapon. It was a fun idea, but I cannot currently recommend this build. I will keep this guide up as a number reference to those who are interested.

Just a test build maximizing crit damage for the beam secondary fire. It will rely on arcane avenger/cat's eye/vigilante set procs for crit chance. I wasn't sure if it would be better numbers to go for [Cat's Eye] or a loaded up sentinel with the Vigilante set so I did the math myself. Riven is fairly conservative and should be easy to brute force, third buff would be damage if you were to try for one. I selected the rough mid way point on the riven rolls to get a number I could work with. It should take 13.5 shots to get to full stacks on primary [Frostbite] and the gun fires 12 a second. My findings on the damage are thus:

Firing cold(no buffs)
248.64base damage from one shot fired=(10.8cold+33.6corrosive)5.6multishot Ratio is roughly 76% corrosive and 24% cold damage
Crit damage is 20x when cold

|normal| |yellow crit| |Orange crit| |Red crit| |Total|

Just 1 piece Vigilante set (2% crit chance with 5% of those being raised one crit level)
243.66+ 97.46+ 99.45+ 0 =440.57 per shot, 5,286.84 DPS

Arcane Avenger and Vigilante 1 piece (47% crit chance with 5% of those being raised one crit level)
131.77+ 2,220.34+ 2,337.21+ 0 =4,689.32 per shot, 56,271.84 DPS

Arcane Avenger and Vigilante 4 piece (47% crit chance with 20% of those being raised one crit level)
131.77+ 1,869.88+ 9,348.86+ 0 =11,350.51 per shot, 136,206.12 DPS

Cat's eyes and 1 piece Vigilante set (62% crit chance with 5% of those being raised one crit level)
94.48+ 2,928.97+ 3,083+ 0 =6,106.45 per shot, 73,277.4 DPS

Arcane Avenger and Cat's eyes and 1 piece Vigilante set (107% crit chance with 5% of those being raised one crit level)
0+ 4,381.4+ 11,208.8+ 6,944 =22,534.2 per shot, 270,410.4 DPS

Firing hot(all buffs engaged)
461.76base damage from one shot fired=(10.8cold+33.6corrosive)10.4multishot Ratio is roughly 76% corrosive and 24% cold damage
Crit damage is 31.5x when hot

|normal| |yellow crit| |Orange crit| |Red crit| |Total|

Just 1 piece Vigilante set (2% crit chance with 5% of those being raised one crit level)
452.52+ 271.97+ 443.74+ 0 =1,168.23 per shot, 14,018.76 DPS

Arcane Avenger and Vigilante 1 piece (47% crit chance with 5% of those being raised one crit level)
244.73+ 6,391.45+ 10,428.15+ 0 =17,064 per shot, 204,768 DPS

Arcane Avenger and Vigilante 4 piece (47% crit chance with 20% of those being raised one crit level)
244.73+ 5,382.27+ 41,712+ 0 =47,339.6 per shot, 568,075.2 DPS

Cat's eyes and 1 piece Vigilante set (62% crit chance with 5% of those being raised one crit level)
175.46+ 8,431.27+ 13,756.28+ 0 =22,362.74 per shot, 268,352 DPS

Arcane Avenger and Cat's eyes and 1 piece Vigilante set (107% crit chance with 5% of those being raised one crit level)
0+ 12646.91+ 114,686.34+ 153,453.11 =280,786.36 per shot, 3,369,436.32 DPS