First, all backdoor hacker does is remove your defense in depth as they will always have a way in to your homeworld.
In a normal game where there is no-exo wave source, I am of the opinion that this is only moderate difficulty.
I'm actually facing one in my current game, but I'm playing a low AIP game with most options off. (AIP is looking to be in the 50 to 60 range when I start my first HW attack.)
That's a very different beast then what you are facing. In a game with exo-waves enabled, defense in depth is mandatory, you simply don't have the time (and forces) to kill the exo-waves before they reach your command station, even if you have every scrap of your forces in your home system.
I'm not going to call a backdoor hacker an impossible challenge to complete a FS campaign with, but surviving those final 30 minutes will be very close to impossible.
As for your current game, I'm not sure I have any advice for you. I'll take a quick peek at the save but you are in a tough spot regardless.
D.
edit: Taking a quick peek at your save, the only suggestion I have is to see if you can find the knowledge for Gravity Turrets. That will buy you more time but I can't say if that will make it winnable.
Hopefully someone with more FS campaign experience will come along, because I prefer low-AIP games, I very rarely play FS myself so I don't have any other suggestions to help out.