Why are 15" guns not considered? As I see it the options are 15(0), 16(0) and 17(+1), right? In this case, especially as you are trying to limit the cost, I would either go with more (8-10) 15" guns for to get more hits quicker or go for the increased punch per gun of the 17" gun.
The 4" Deck Armor of the Variant B ships is a bit light for my taste, though I agree with your general reasoning on armor.
So just out of curiosity, what would be the tonnage/cost of a Variant B with 4,5" Deck and 3x3 (or 424 KGV layout) 15" guns, everything else being equal?
Because they are worst. I have just estimated it in my mind. But I can try do correct calculation.
Immunity zone ranges
First, I will start that everything is based on immunity zones. So to ships firing against each other have immunity zones against enemy ships and it is range. Putting together both ranges we are getting 4 combination of ranges. For simplification to show principle I consider immunity zone for citadel same as for turrets. These zones are:
Zone A range - our ship outside immunity zone, enemy ship inside immunity zone - this range needs to be avoided at all costs
Zone B range - our ship outside immunity zone, enemy ship outside immunity zone - this range needs to be avoided too as we consider ship against overhelming advantage
Zone C range - our ship inside immunity zone, enemy ship inside immunity zone - this range needs to be avoided too as it is matter of attrition and we will loose because of overhelming advantage of enemy
Zone C range - our ship inside immunity zone, enemy ship outside immunity zone - this is exactly we want to be!
But C range is usually very small. If it is just around 1000-3000 yards, it is difficult to be within and it means that still a lot of our hits do not penetrate enemy armour. So we need to increase such range as much as possible. How we can do it? There is 2 possibilities:
1. increase our immunity zone at range where enemy has no immunity zone - this can be done by adding armour but it is difficult to predict what way immunity zone need to be enlarged so if horizontal or vertical armour should be thicked
2. decrease enemy immunity zone at range where we have immunity zone - this can be done by increasing caliber of our guns, it works both way against horizontal and vertical armour
Comparison 15" vs 16" guns using immunity zone principles
Now we can look at design comparison. I will use B1 design as a base. I will not correct X turret, I will use X turret instead of Y for all version to eliminate issue for comparison.
Variant B1 - 34100 tons, 27 knots, 3x2x16", 128M, 13" inclined belt, 4" deck, 15" turret faces, 6.5" turret tops, 4" CT, 2" secondaries, TDS lvl 4, 14x5" casemates guns, 10x4" DP guns, 16 LAA, 24 MAA
We will use this Variant B1 against ship with 15" guns. So we need to calculate immunity zone of B1 vs. 15" guns. Than we need to adjust B1 to B15 variant with 3x2x15" guns and same immunity zone vs. 16" guns as B1 vs. 15" guns. As turret tops are inclined and we do not now exact effect I would take 90 % of turrets top armour as effective armour for horizontal protection.
B1 immunity zone against 15" guns:
- citadel 12500 - 23000 yards
- turrets 11350 - 27500 yards
No we need to calculate how much armour is needed for this immunity zone against 16" guns
belt: 15.9" ==> 14.5" inclined
deck: 4"
turret armour: 16.5"
turret top: efficient 5.95" ==> turret top 6.6" rounded to 6.5"
Now we will design B15 variant with 3x2x15" guns and this calculated armour so both ships has same immunity zone against each other.
Variant B15 - 35000 tons, 27 knots, 3x2x15", 131M, 14.5" inclined belt, 4" deck, 16.5" turret faces, 6.5" turret tops, 4" CT, 2" secondaries, TDS lvl 4, 14x5" casemates guns, 10x4" DP guns, 16 LAA, 24 MAA
Is it all and can we compare costs? No, it is not. Both ships has same immunity zones, but we need to look at rate of fire and damage potential. Rate of fire is the same so only difference is damage by one shell.
Damage of 16" shell vs. 15" shell is calculated as: (<weight 16" shell>/<weight 15" shell>)^(0.66)
Weight of 16" shell: 3000 lbs
Weight of 15" shell: 2048 lbs
Ratio is: 1.14 x 15" shell is equal to 1 x 16" shell
Conclusion:
Version with 16" guns vs. 15" with same immunity zone vs. each other is 2% cheaper and has 14 % higher damage potential. So it is clear winner.
Deck armour
Is quite heavy and did not do too much against 16" shell. Increasing deck armour about 0.5" means increased immunity zone only by 1000 yards.
For example variant B1a with just increased deck armour to 4.5" needs 35200 tons and costs 132M so 3% more.
If I take your example and create B1b variant with 4.5" deck armour and 10x15" guns using KGV layout as it weights less, I need 43700 tons, 164M!
Compare to variant B4 with 8x17" guns which has still 4 % higher broadside than B1b. As you can see both costs are about equal 163M for B4 vs. 165M for B1b. However immunity zone is completely different level. Immunity zone of B4 is same as B1 above, 9500 yards for citadel and 16000 yards for turrets. On opposite B1b immunity zone against 17" guns of B4 is this:
citadel: 18500 - 23100
turrets: 17300 - 26000
It is much worse!
Conclusion:
16" gusn are always better than 15" guns. The same is 17" over 16". However there is question of minimual number of guns remaining.
Note:
I have not taken into account bomb damage however in 30s bomb damage would not be significant and later these ships will be not the backbone of the navy and can still withstand some penetrating hits. And to have really protection against heavy AP bombs the deck armour would need to be heaiver at least 1-1.5" which is quitet a lot.