That's really odd. If you can't get it unclogged, then I believe it's the entire plate, call Agilent to see if they sell it, then go online to see if you can find a column heater for less that you can use for parts. If you had a newer column compartment, you could use the low-volume heaters that mount directly to the left or right plates, but I think that's probably not an option for you. The part number is G1316-60006, and a short search online revealed prices is the $15-1700 range from non-Agilent suppliers, and $1437.36 from Agilent.
A couple questions:
1. What type of tubing are you running from the autosampler to the column compartment - PEEK or stainless steel? If you're using PEEK, I'm wondering if there isn't a small piece lodged in the entrance to the heater, that only affects the system when you reconnect tubing to the entrance because it then blocks the entrance?
2. How did this problem present itself? All of a sudden, right after an injection, in the middle of a run? What mobile phase were you running when it happened? What were you injecting (sample type, sample diluent)? Some details might help, might not.
I would attach a line directly from the pump (no point in subjecting the valve to unnecessarily high pressures) to the unclogged heater and then to the entrance port of the clogged heater, and put both heaters at about 60 C. Then I would increase the flow rate in 50 to 100 microliter per minute increments, using isopropanol. Make sure to attach some wide-bore PTFE tubing from the exit port of the clogged heater and direct the flow to a waste container. I would do this just to see if there is a flow rate that may (in combination with a little heat and time and associated pressure) alleviate your issue. Might help, might not, but it may save you time and a thousand dollars. Just make sure to record the backpressure through the system to just before the clogged compartment prior to trying this flush procedure, so you can determine the backpressure developed by the clogged heater.
If you can't get any flow through the heater, you can always use the 3uL heater - I never saw much of a difference between using the left or the right, and I doubt you will either unless you're running at high temps (50 C and up) or high flow rates (more than a mL/min or so). You can also remove the 6uL heater and try to visually inspect the entrance to see if there is a blockage, if you can't see inside it well enough with it installed.