Yes I understand it quite well, ~20% scale up which is 20% longer, and 20% wider is quite a big change in mass.
The big problem is either staging or being able to shift it to a shallow enough trajectory and get enough velocity, but it might be solvable if you can do a zoom climb type launch from an aircraft similar to how Pegasus/SpaceShip One do it.
https://en.wikipedia.org/wiki/Pegasus_(rocket)
How is a 20% scale up going to get you anywhere close to the extra dv needed? 20% on each dimension is still only 70% more rocket, where you need something more like 10x more rocket to make that leap.
That's 1.2^3, or 70% more... not even close. Also remember that making the rocket bigger also increases the dry mass, which further increases the fuel requirements.
The difference in ISP between a liquid-fueled rocket (say 450s for LH2/LO2, 350s for RP1/LO2) and solid propellant (250s) makes a huge difference to your prop fraction. To get, say 8km/s delta-v, just plug it into the rocket equation:
