You don't start with a dog and there is a negative response to meeting the dog so you are not forced to take the dog. But hey lets not let logic and facts ruin the narrative you want to tell which is i am a poor gamer unfairly restricted by Bethesda.
Having a spouse and a Child is not a restictive background. There is nothing inherent to this background with regards to the type of person your character is. You can be...
A conartist using the marriage to swindle your spouse out of their savings
A serial killer using the marriage and kid as a beard to appear normal and defelect suspision.
A spy using the marriage to blend in while you report back to the chinese government.
A thief that also happens to have a family. You steel to provide for your family.
A disgrunteled unhappily married person thinking of leaving but... the bombs fell.
A soldier who hates this life and wishes only to return to combat where you feel you can serve your country
A reformed gang member who is trying to turn their life around.
A straight up suburban drone living the 50's dream
A crooked cop using the system to gain as much wealth before you run off with your mistress
A person who suffered a complete breakdown when you awoke in a vault filled with dead people so you imagined this perfect life before the bombs fell to give you comfort in this terrible radiation blasted distopia.
This is just 10 quick wildly divergent possible characters that work with zero effort.
The most limiting aspect of your character is not the wife, the kid or the voice actors but that you were born before the bombs dropped. As this limits any background to what was plausible pre war. You (the PC) seem to have zero knowledge of the world post war. So any background referencing post war is out. unless you do the I have amnesia trope.