I think we can all agree that the crux of this question is in the New Shepard specifications, feel free to offer improvements to these. Here are the ones I arrived at (note: there is no recovery of the booster in this analysis):
This reddit thread (that is now 5 yrs old) contains some estimates of tank volume and empty booster mass and I ...
WEll my answer is yes, it- "could" be done
perhaps adding 2 Solid boosters in a C-Slot configuration on the sides to give it enough power to go sideways
then do the russian/titan hot staging technique with a high altitude/vacuum solid booster designed to burn for a long time to reach orbit and is adjustable as to meet the specific payload ...
Note many other motors in the catalog have those features (non exhaustive list: STAR 13B, STAR 17A, STAR 27H)
They are igniters as shown in this drawing of the STAR 27H from The IBEX Flight Segment
The other motor presumably has an igniter mounted at the other end of the case - that may be the fitting barely visible at the left.
The squat end of the spectrum has little to do with solids versus liquids and everything to do with aerodynamics. Spherical tankage is most weight-efficient, so you'd expect squat stages in cases where aerodynamics don't dominate such as your Mars Ascent Vehicle (flying where atmospheric drag is on the order of 1% what it is for Earth ascent) or the Apollo ...