Something similar happened to me. Mine worked fine for a week. I have one bass with active electronics and one with passive. One day I took it off the charger (it was solid green), plugged it into my bass with active electronics and it didn't light up. Not knowing what the issue was, I put it back on the charger and it no longer lights up, though the charger blinks red like yours. Also, the transmitter was extremely hot to the touch for about 30 minutes after even though it wasnt plugged into anything.
My issue seems similar to yours ,though my transmitter doesn't light at all and yours at least blinks alittle. The good news is that I opened a ticket and they are sending me a replacement. Wondering if there is an issue with the batteries they are putting in these G10s