Unless there is a conflict with your bull terrier there is no reason at all why the new spaniel should not live in the family home once it is fully mature.
A small spaniel puppy will generally be very exuberant and probably test your existing pet to the limit. To maintain harmony it is essential the introduction is done carefully on neutral territory and subsequent progress carefully monitored.
As the spaniel is to undergo thorough training for the gun, however, it may be more practical to keep it in a kennel away from any possible conflict in the home.
Your children must also realise the new addition needs its own free time and games are probably not conducive to the development of a gundog – especially tug-of-war!
Any puppy needs its own space in which to feel secure, whether a kennel or a separate room in the house. You, as the trainer, will need to be the focus of its attention and this aspect is easier to develop away from distractions caused by family life.