The ~30 mil core with build-up 7-10 mil prepreg stackup is more common, AFAIK. JLC's 5 mil (or less, whatever it is?) prepreg is a somewhat less common / newer sort of thing. So, I would target the first one, and make it clear in the notes that it's the stackup expected, and note critical nets if applicable.
Similarly for 6 layer, I think 7 mil prepreg is the usual go-to (or was), so whatever that stackup comes out to.
Helps to have a txt file in the ZIP making this clear, as well as on the drill drawing or whatever. I usually start with something like:
Board Details
-----------------
Number of Layers: 4
Board Thickness: 0.062 in
Board Material: FR-4, Tg >= 140C
Board Size X: 4.00 in
Board Size Y: 4.00 in
Copper Weight Outer: 1 oz
Copper Weight Inner: N/A
Soldermask Color: Green
Silkscreen Color: White
Plating: Lead-Free HASL
Number of Sides requiring Assembly: 2
Approximate Number of Pads, Vias, Holes: 1337
Approximate Number of SMT Components, Top: 69
Approximate Number of SMT Components, Bottom: 0
Approximate Number of Thru-Hole Components: 42
Approximate Number of BGA/QFN/Leadless Components: 1
Blind & Buried Vias: 0
And add any controlled impedance notes at the bottom say, or recommended/required stackup with the geometry specs there. Can show the stackup on the drill drawing with a table, or graphically.
You can always custom order, of course, at the expense of increased cost and lead time. It's not too bad, like, I've ordered a 20-20-20 mil stackup from PCBCART I think it was, for a high power inverter (hence the increased spacing), which was like $250 for, 4 or 5, or was it 10, I forget?
And that's just protos, the incremental cost is basically negligible once you're in production.
Tim