I couldn't sleep so I made spreadsheets instead. I compared a bunch of '36 heavy hulls with engine 2, and all different combinations of homogenous gun tier and count, and different levels of armour against SHBB with engine 2 and its different gun counts. No other equipment on either ship. I used your armour mechanics, (piercing/armour)^2, and the speed which was friendly/enemy. I didn't use any designers or admirals or ship XP, this is all basically just theoretical ship building right now to test the mechanics and to guide us forward with balance changes we would have to make if these mechanics were accepted.
Here is the google sheet. It is hard to read because I was kinda all over the place and I tend to write things in codes that only make sense to me at the particular time I wrote them. The first two sheets don't really matter though they do have intersting information, focus on the BB2/E2 Comparison.
The BB2 and the SHBB seemed to prefer to be mounted with 4 gun batteries, it offered the best balance between cost, speed, and raw firepower. SHBB could get away with 3 or maybe even 2 guns depending on the specific configuration of the enemy ship, so you can start there and then refit on some guns later as the enemy develops.
The SHBB having more penetration and armour, and more basic firepower was able to at basically all levels provide significantly more damage per IC than the BB with any tier and amount of guns. The BB2 had a lowest cost per heavy attack of 235 IC, using 4 tier 4 guns against a SHBB with 5 guns to slow it down. A 5 gun SHBB shooting at a 4 gun BB2 is only spending 68, 53, or 88 IC per attack, depending on the tier of armour the BB2 was using. That 5gun SHBB cost ~20k IC before the +50% you mention, and it'll 2-shot the BB2. The BB2 costs 14160 IC and would need 13 hits to sink the SHBB. These comparisons seem to heavily favour the SHBB.
All of the profiles of "usable" ships ended up being above 80, accuracy was the same all around. TI doctrine is also basically a waste if you wanted protection against heavy attack, the profiles are pretty terrible when you're mounting 3+ guns. You'd have to stack a bunch of other modifiers like bold/concealment expert on your admiral, and raiding designer though I suggest coastal to save time/resources ( or just have more ships)
I think armour/piercing is being weighted too heavily because it is squared. I've also ignored the combination of factors that could end up doing as much as doubling armour values, which makes having piercing extremely important.
The speed mechanics don't really have that big of an impact. Maybe square those too if you wanted to bring them more into the center stage, but as you could see the from speed comparison page (which is using a BB1, which is pretty slow) the differences in speed aren't really all that big most of the time, even squaring them won't make a huge difference. except in edge cases where we got a zoomy boat with 1 gun facing off against a super heavy laden down with all of the guns on earth, but those aren't really what we're looking for, I'd imagine.
Of course, other things we could shift around here are the amount of piercing/armour the different modules have, the speed modifiers of the modules, the attacks the modules offer, and just the IC and resource costs. But those are things that we could change to bring the simulation more in line with what you want, without having to add fancy new mechanics.