I went with over the pins between the FPGA, IT6613E and the HDMI socket with a solder iron and solder wick and reflowed the ESD protection on the HFMi traces again. This seems to resolve the problem with the digital noise.
Now I only got a problem with the OSSC not beeing able to flash the serial flash and loading the profile settings. JTAG flashing works, but maybe I have an incompatibility with the SST25VF016B I use as a replacement for the hard to get original serial flash.
But this problem here is solved now. Thanks a lot!