Apart from your ascent profile, there's nothing keeping you from getting into orbit with this craft.
I made a reconstruction of your design and got it in a 100x100 km orbit with ease and quite a bit of fuel to spare, you can even keep the RAPIER in 'automatic switching' mode:
Ascent profile:
1. SAS on, launch and start at a 30° attitude angle (no need to touch the controls after setting the initial angle)
2. When your speed is 1200 m/s, set SAS to 'prograde'
3. As soon as the RAPIER switches to closed cycle, set SAS to 'stability assist'
4. Burn to desired apoapsis height, cut engines and set SAS to 'prograde'
5. Circularise orbit @ ~apoapsis
Give it a shot and see if you can get to orbit, please let us know if you're successful (or not)!
Yakuzi's post in WHy is my SSTO not getting speed was marked as the answer