If the solenoid has been in service for a long time period the disc and contacts wear away eventually and will cause the same symptoms you're describing. You might only need a solenoid and that can be verified by disassembling the solenoid, and checking the disc and contacts inside.
If the solenoid isn't the problem I myself would rebuild my own core, but that's my preference. ''
Among the Rock Auto offerings I'd likely go for the new starter, cores that get rebuilt vary greatly in quality, as do rebuilders. In my own vehicles I always try to buy new over reman every time. Diesel trucks I own I buy new starters because of the extreme loads they're subject to. Pontiacs are also hard to turn over, and always have been, just by design they are harder on starters than most other GM V8s.
Also with remans you have to send them your core, extra work and expense.