You could use the railgun to cheat a little; at a modest 3 gravities, that
60 km railgun would get you up to about 2 km/sec. Not sure how much that
would cut your cost-to-orbit, but it would probably be a significant
amount.

2000 m/s muzzle velocity looks a bit low to me. I like 2500 m/s.

Here are the results from a simulation of various single-stage propane-LOX
rockets fired from guns at the equator. The payload masses rise with launch
velocity so fast because the thing being held constant is the engine thrust.

payload_mass delta_v drag_loss gravity_loss throw_mass muzzle_vel gun_elev
kg m/s m/s m/s kg m/s degrees
1803.00 1600.00 2086.93 497.44 3576.36 8223.86 12.05
1314.63 1900.00 2073.22 515.91 2870.77 7931.33 13.73
996.89 2200.00 2016.40 535.96 2398.07 7597.79 15.42
776.89 2500.00 1929.73 557.93 2060.36 7236.58 17.16
618.42 2800.00 1821.94 582.01 1809.70 6856.80 18.97
501.27 3100.00 1700.54 608.29 1619.85 6466.46 20.87
411.62 3400.00 1570.54 637.28 1470.42 6070.84 22.88
341.88 3700.00 1436.82 669.24 1351.50 5675.41 25.02
286.69 4000.00 1299.67 704.56 1255.60 5281.03 27.33
241.92 4300.00 1171.67 744.15 1175.64 4900.97 29.85
205.70 4600.00 1045.85 788.06 1110.65 4529.21 32.59
175.91 4900.00 927.11 837.17 1056.97 4171.61 35.60
151.13 5200.00 816.74 892.42 1012.39 3830.68 38.91
130.30 5500.00 715.68 955.04 975.15 3508.91 42.59
112.83 5800.00 625.00 1025.58 945.31 3208.76 46.65
97.97 6100.00 544.87 1105.86 921.24 2932.44 51.15
85.28 6400.00 475.52 1197.42 902.49 2682.24 56.11
74.43 6700.00 416.92 1301.54 889.08 2459.81 61.51
65.07 7000.00 368.78 1420.52 880.35 2267.05 67.33
57.00 7300.00 329.92 1555.42 876.67 2103.48 73.43
50.01 7600.00 300.71 1707.65 878.01 1969.71 79.65
43.90 7900.00 280.17 1878.94 884.15 1864.60 85.76

This is effectively a two stage to orbit design, with the gun acting as
the first stage. There are a number of constants for this simulation:

thrust2mass 500.00 m/s^2 tank2fuel 20.00 m -
exhaust_v 3300.00 m/s motor_thrust 5000.00 N
pointing 1000.00 m - fuel_density 1222.00 kg/m^3
fuel_price 7000.00 m $/kg length2diam 10.00 -
cd 150.00 m - collar_mass_fraction 200.00 m -
final_orbit_alt 360.00 K m muzzle_alt 0.00 m
orbit_vel 7692.43 m/s

Like most orbital insertion rockets, this one has a short high
acceleration stage (the gun) followed by a long low acceleration
stage. The short high acceleration stage is not a cheat -- it
accomplishes three important objectives:

(a) it gets the upper stage into thin air, where a high-expansion
high-ISP LOW-PRESSURE engine can operate (no turbopumps),
(b) it eliminates the need for an upper stage that can lift its own
weight, and
(c) it sharply reduces the gravity losses from a low-acceleration
upper stage.

Guns are particularly nice for first stages, since they have gigantic
reaction masses that give them high energy efficiency. I like simple
chemical guns, either gunpowder or maybe LOX-propane, as they are
generally reusable and don't have anything as expensive and development-
intensive as turbopumps or railguns. They also won't deliver muzzle
velocities over 3 km/s, and 2500 m/s is a bit of a stretch.

My particular favorite above is the delta-v=6700, muzzle_v=2460 point.
Note that the rocket, when it lights up, masses 889 kg and has 5000 N
of thrust. That's just over a half G of acceleration -- the thing
loses speed for a good chunk of its flight, and gets to orbit anyway.

Nothing like this is ever going to be man rated or even useful for
most satellites. With some ingenuity, you might be able to launch
some sturdy satellite bits -- perhaps the RCS system and fuel, and
perhaps the solar arrays, folded up to fit inside the propane fuel
tank of the launched rocket, with a foamed silicon carbide backing to
give it neutral buoyancy to survive the launch.