1200G tonearm with stock headshell has an effective mass of 12g. Without the headshell, effective mass is approximately 4.5g. If you install your 15g Ortofon headshell, you'll have a tonearm which will be approaching 20g in effective mass. You should be able to make that work with Lyra. I don't see why you're having issues there.
Having said that, Lyra in general is an obnoxiously bright cartridge and the stock tonearm on the 1200G is just adequate, but nothing special.
You can try other tonearms, arm boards, and cartridges, but now you're investing significant money into the 1200G on top of the $4000 retail price.
There are better options such as a Gyro SE with an SME tonearm, or entry level SME tables like the Model 10.