It doesn’t work like that since this is on a galactic scale, the only thing you need for a galaxy to be 350 trillion times as luminous as the sun is a galaxy with 350 trillion stars which are just as luminous as the sun.

Form an intra galactic viewpoint it wouldn’t look much different (at least to a human observer) and the only luminosity that you would really care about would still be your local star unless you are really close to something special that can cook you.

Now, if you want to know what it would look like if the Sun suddenly became 350 trillion times more luminous, I think would probably look like your entire body melting/evaporating in seconds.

I think it would be faster than that. Normal luminosity is roughly one kilowatt per square meter. This would be roughly 3.5e17 W/m^s. The largest nuclear bomb ever detonated was around 2e16 joules, so it would be like getting a Tsar Bomba to the face ten times a second.