He could roll it on its side. The vertical LPD axis would then line-up with the horizon. He might still have azimuth errors, but he'd be in orbit and could finesse it from there, fuel allowing.
It might work. You'd have to estimate and control the angle between the LPD axis and the horizon, perhaps by using the small horizontal scale markings on the LPD; not sure what they were for. (Actually, this particular maneuver would be yaw, not roll. The LM axes were defined with respect to the astronauts' heads when in their flight positions, not with respect to the primary thrust axis.)
However, it would still be really important to control your launch azimuth. Otherwise you might well get into a stable orbit only to discover that your orbital plane is so far from the CSM's that you can't rendezvous with the available fuel. Even small plane changes can be very expensive in fuel, which is why the latitude of the launch site, the flight azimuth, and the launch time are all so important.
Perhaps you could do both by frequently yawing between heads down and heads to the side, or maybe even doing a continuous yaw that shows you the lunar surface, one horizon, the earth and the other horizon in sequence through the front windows.
The problem is that without a guidance system you'll have to continually compensate for the thrust vector not going directly through your center of mass. Note how even with a functioning guidance system all the LMs "wallowed" quite rapidly during ascent as the RCS engines were fired to compensate for this small unwanted torque from off-axis APS thrust. Come to think of it, a continuous yaw might help this problem too. Many satellite kick motors are fired with the satellite rolling around the thrust axis specifically to cancel out off-axis thrust.
