Executive summary: the barrage line is pretty weak.
While most city-attacking pults will get city raider, I tend to give the first couple of units the collateral promotions. The odds are very high that you are going to at least lose the first couple of siege units if not more so, IMO, might as well do the most damage with them and collateral appears to do just that. Increases the survival chances as well for the following city raider pult attacks.
I don't believe the gains are very much, in practice. You're really only going to see a significant gain if the extra collateral damage pushes you over a jump point.
Assuming my old collateral calculator is still relevant....
vs Archers
12 collateral damage (base)
13 with Barrage I
15 with Barrage II
vs Longbows
9 collateral damage (base)
10 with Barrage I
10 with Barrage II
If you then take those numbers over to the combat calculator, the news doesn't look very good. Giving an Archer 115% bonus on defense (135% less the 20% for city raider I), you can scan down the attacker win percentage column.
For example, 12 damage will give the next CRI catapult a 35.7% chance of winning; 13 damage will produce 36.4%.
Two base catapults will do 12 x 2 = 24 collateral damage, so the third CRI cat has 44.6% winning percentage. Two barrage I catapults will do 13 x 2 = 26 collateral damage, the third has 46.13% chance of surviving.
Under those same circumstances, City Raider I takes survival chances of the first catapult from 8.8% to 18.5% - so it looks clear to me that you'll save more catapults in the long run by promoting on the CR line right away.
Now, 135% probably isn't a good number to choose - I grabbed it because it was already loaded in the sheet. You should go through the exercise with whatever defensive numbers you think are appropriate.
The key points to remember are (a) that defender bonuses don't change collateral damage, (b) you get significant differences in survival odds when you cross a jump point, and (c) if you do any damage to the top defender, it's going to be more than the collateral damage, so you can ignore that defender further
Side comment: a complete analysis would also work in the odds that you don't injure the top defender at all.