What lower-profile options do I have to replace a triple 2x12 beam?
You can do it with LVL.
Grabbing the first LVL spec sheet I came across (Boise Cascade, from https://www.bc.com/versa-lam-lvl-span-size-chart/), I find a Versa-Lam LVL 2.1E 3100 has an allowable bending stress of 3100 psi. That will get you above the 2930 psi from your friend's calculations. So let's start there.
That LVL comes in (among other sizes) 3.5&quot; x 7.25&quot;, which is really close to your existing doubled 2x8. In fact, it's enough bigger that it'll give you I (=bh3/12)) of 111.1 in4, up from the 90.4 in4 of your current beam.
Bending
So then your bending stress (fb=MC/I) becomes (74390 * 3.625)/111.1 = 2426.2 psi. That's below the 2930 psi allowable, so that will work.
Shear
The allowable shear for that LVL is 4821 pounds -- well above the 1725 pounds that you need to carry.
Deflection
The LVL also has a higher modulus of elasticity (E) than typical dimensional lumber, but will have a smaller I than the triple 2x12, so let's check.
Deflection = 5WL4/384EI = 5 * 20 * 172.54 / 384 * 2,000,000 * 111.1 = 1.04 inches. That's slightly over the 0.96 inches allowed. You could go up one size (to a 3.5 X 9.25 or to a 5.25 x 7.25) and that would get the deflection back under 0.96 inches.