Looking for a routine to correct for Local Standard of Rest (LSR)
There are several corrections here, and as you suspect, all of them can be done in astropy.  You'll need to do something like this:

Transform your observed velocity to the solar system barycenter (thus correcting for Earth's rotation on its axis, and its orbit around the Sun.  There is an example here in the astropy docs that shows how to do that.  If you follow that example, you'll first need to define your radial velocity as an astropy &quot;quantity&quot; object with units, e.g. something like this:

import astropy.units as u
# Define RV of 30 km/s
rv = 30 * u.km / u.s  

You can ignore the discussion in that example of the relativistic correction, and just add the barycentric correction to your velocity, e.g. corrected_rv = rv + vcorr

Then, you can correct for the fact that the Sun isn't moving quite in step with the average of the nearby stars, and convert from the barycentric radial velocity to the local standard of rest, or LSR.  Astropy has some examples here but they don't do anything quite like this, so here's some example code.

Set up an observation that has the coordinates you observed, and the corrected radial velocity from step 1.  (I'm continuing with 30 km/s here, but you would substitute whatever you got in step 1.)
from astropy.coordinates import ICRS, LSR
my_observation = ICRS(ra=17.75*15*u.deg, dec=-29*u.deg, \
     pm_ra_cosdec=0*u.mas/u.yr, pm_dec=0*u.mas/u.yr, \
     radial_velocity=30*u.km/u.s, distance = 1*u.pc)

For specificity, I've chosen coordinates that are toward the Galactic Center. Here we're specifying zero for the proper motion terms since astropy won't do the velocity transformation if it doesn't have all three velocity components; that's also why we need the distance.  (You can experiment and convince yourself that the values don't matter in terms of what you get for the radial velocity.)  Now, you can transform that to the LSR frame and print out the new radial velocity:
new_rv = my_observation.transform_to(LSR()).radial_velocity
print(new_rv)

which gives 41 km/s, since the motion of the Sun relative to the LSR has a component of 11 km/s (the &quot;V&quot; component in the UVW system), for example in this derivation of the solar motion from Gaia data.