Keep it running and you have a better chance, if the filter stops the bacteria starts to die off within hours. If you can get some ammonia to dose the tank with till you get your fish then all the better. Otherwise just keep the filter running and restock nice and slowly, almost like you're starting again and do water tests to make sure you're not getting a mini cycle as a result. Do you have to euthanise today? could it wait till Friday? Not an easy question when you have a suffering fish, but it would make you're transition easier if you can leave it a day or two.
What's the fish dying of btw?
The goldfish to be honest im not sure what it is.
I have a thread in the cold water section where it started.
It seemed like possible fin rot, I left it for a day. Wake up the following morning and half his tail is missing!
It was suggested that it was to fast for fin rot so maybe bordem?
few days later he moved into my cycled 15gal, added melafix and he improved, all the redness in the tail fin was gone. happy and normal.
This Saturday i had to go away till today so i explained to family how to look after him (not alot to do really)
ive come home to find his top fin is all but gone, his bottom is almost gone,his dorsal is bright red, I -think- one of his gills has a lump, theres a white ring around the later part of his body and I found him laying on the filter out flow tube.
I honestly cant seem him making it and it pains me to even see him like this

Thankfully I have some pure clove oil so it will be a painless sleep..
The only other fish I have is a male betta i could move into it but it will take awhiel for the water to get hotter and I need to find a way to lower the current for him.