Normally the loop loss on a broadband circuit should remain stable. A fluctuation of 3dB could be a sign of a poor joint between your property and the exchange . You mentioned that you were told there was a line fault when you first reported the problem to EE. If this was a telephony fault (battery contact, rectified loop ,etc.) you should have been referred to your telephony provider. However the line has to be good enough for broadband. As part of the of the BT Wholesale Knowledge Based Diagnostics available to ISPs an Enhanced Copper Line Test would have been run and it is possible this failed - the fault code is CU34. If this happened then a fault should have been raised into BTW and ultimately you would be offered an appointment for a PSTN engineer to bring the line up to standard for broadband. However if the condition affecting the line is intermittent is quite possible to get a pass on a subsequent test.
Given that you were getting about 3meg it is almost certain that you will also have an LTB - Lower Threshhold Breach fault. When you broadband is first provided after the first 10 days benchmarks are set for a Maximum Stable Rate (highest consistent downstream sync speed) and a Fault Threshhold Rate (80% of the MSR). If your line rate drops below the FTR you have an LTB and if the cause cannot be proved to customer kit/set-up then a fault should be raised in to BTW by the ISP.
EE does not apply capping or banding. Banding is an automatic function of Dynamic Line Managemet built into the broadband equipment at the exchange. BTW will in some rare circumstances cap what they call a flapping line - this is usually as a last resort on a very unstable circuit.
If DLM has applied a banded profile to your circuit it is because it has been triggered to do it - should your circuit become stable of its own accord DLM will act in reverse to give you back the available bandwidth but it can be very slow to do this.
There is no point in trying to force EE to override DLM unless it is absolutely certain that the root cause of the instability affecting your circuit has been identified and eliminated.
I know it is difficult but be persistent with the EE Tech Support Team - if they tell you there is a line fault ask them what the actual fault is, what the KBD output code is and what they intend to do about it.
Moving to another ISP and staying on the same type of broadband technology without identifying what is causing the low sync speed will just result in you taking the fault with you.
"REMEMBER - If you do get trapped in your flat, try not to get trapped in your flat." Vic Reeves