I do not think the balance arm is the problem. You can test this easy by not using it at all. Put a 12 inch lp on the holder. Use your hand to hold the lp like the balance arm would and start the unit to auto.
I think but do not know this bsr table that good but if like there latter tables. Under the platter is a gear that would move the arm. I think the gear is jammed. In this video you can see this gear.http://www.youtube.com/watch?v=P2H9BZmsrjs
Edit I am wrong the gear on this is under the unit. Service guide here.http://www.radio-workshop.co.uk/service/bsr-monarch.pdf
Troy, the issue with this BSR Monarch is most likely a gummed-up metal pin buried in the mechanism underneath.
Unless you know what to look for, (as us ahem, experienced techs do) this pin and it's function during the change cycle will baffle anyone - and thus create a never-ending thread (as mentioned in my previous post).