Is this a roof top unit on a travel trailer?

If the cap made the compressor run, how long did it run before blowing the cap? It sounds like the compressor needs a hard start kit along with a new capacitor.

The compressor may have been ok, or may have a weak motor, but if it runs, it runs. If the valves are bad, and it doesn't compress, the compressor is bad. Replace the cap first, and add a hard start kit. If this blows, then onsider replacing the compressor. Many compressors have a 5 year warrantee, so the cost is in labor only. Depending on your area, $550 sounds like a warrantee compressor. The price for labor sounds ok.

13.5 duotherm on a 25' motorhome basically just like a window unit.compressor will still start but will only kick on for about 3 minutes then it kicks out. then about 5 minutes it kick back on for 3 minutes. the lines start to get cold when it kicks out. it is full of r-22 it only holds 24 oz. the compressor get warm when it cycles but not hot nor can you smell anything burnt

So you probably have a bad capacitor or overload. Use a 3-1 kit or a 3-1 and a hard start.