Questions comparing damage calculations like this can often be solved with basic algebra, systems of inequalities specifically.
Your normal expected damage per hit is
- \$x\;-\$ %chance to hit (you know, AC-attack bonus). \$x\geq.5\$ (because of nat 20s) and \$x\leq.95\$ (because of nat 1s).
- \$y\;-\$ average damage. \$y\geq1\$.
$$d = xy$$
With Weapon Specialization, that is
$$d = x (y+2)$$
With Improved Critical, if \$z\$ is your crit range, then you do normal damage plus a \$z\$ chance of normal damage again (barring non-doubling stuff like SA and other feat shenanigans)
$$d = x (y + zxy)$$
So if you want to know when IC is better than WS, you are solving the inequality
x (y + zxy) &> x (y + 2) \\
y + zxy &> y + 2 \\
zxy &> 2
So if your chance to hit times your crit range times your damage is greater than 2, then IC is better (this stands to reason, right — you get +2 from WS, so when do you get more than +2 from IC? When you hit, and when you confirm, and the resulting expected damage is larger than 2).
In your example you have a 55% chance to hit, and a 15% crit chance, and an average damage of 14.5 (though you didn't provide information as to where those other dice come from, it's likely they're not actually doubled on a crit because that's pretty rare unless it's making the weapon bigger or something). But just to take it at face value,
\$.55 \times .15 \times 14.5 = 1.196\$, so Weapon Specialization is better.
After your edits, the same formula works — the confirm chance is \$x+.1\$ because of your +2 to confirm crits, and you can just ignore the non-doubling part of the damage because it's a flat addition to either scenario. So you're looking at \$z(x+.1)y>2\$ where \$z\$ is still .15, \$x\$ is still .55, and \$y\$ is now 5.5. So WS is now even more way better in your example.
With this formula you can now trivially run other scenarios. What if you're going to get some power next level that lets you autoconfirm crits? Well, then you just need \$zy>2\$ because the confirm chance becomes 1, right. But with that damage (\$.15 \times 5.5\$), IC is still worse than WS even if you auto-confirm. Let's say you take Improved Critical and get that .15 threat range up to .30 (15-20). Nope, \$.3 \times 5.5\$ is still less than the +2 you get from WS.
Any time you want to see a hit/damage scenario 1 vs. scenario 2, set up a system of inequalities, reduce it to the fundamental variables, and reduce, and you'll end up with a simple formula that will tell you under what circumstances one is better than the other without Anydice or spreadsheets or whatnot.