Hi aafbd and welcome to the WAB forum.
These dilemmas can be difficult to resolve, and especially so without proper examination of the 'patient' ... so don't be surprised if your query is responded to by WAB members with widely differing perspectives.
My response is of course a personal one from my viewpoint and experience as a practising wildlife rehabilitator, so I'll try to stick to providing useful guidence based upon what you have told us so far.
First of all, in situations such as this where a small mammal has been attacked by a cat, the best course of action is usually to keep the patient in a warm, secure and quiet place for a few hours before seeking veterinary assistance ... unless it is bleeding profusely, in which case it should have immediate veterinary first aid.
Because cat saliva invariably contains bacteria that can infect the victim's system to cause fatal blood poisoning, a normal safety precaution is to have a sympathetic vet administer an antibiotic injection. Without this, any 'catted' animal may not survive the first 48hrs following an attack.
If as you say the mouse is still active although the first attack was more than 2 days ago (you probably can't really be 100% certain that it was the same mouse that was attacked both times) then it is possible that its immune system is strong enough to counter any infection ... or it could still be harbouring an infection that the mouse in its state of alarm is hiding from you when you look in on it. Whatever the case though, it may already be too late to now administer effective antibiotics.
Whether or not the mouse will/can survive in the wild with its current loss of a limb is difficult to say without a close examination of the injury.
Apart from using its fore-limbs for running, climbing and burrowing, the mouse needs both to properly manipulate some foodstuffs and for essential grooming to keep its fur clean and waterproofed. Without one forepaw, the mouse will be severly constrained in my opinion, and in such circumstances I would normally advise keeping it in sanctuary.
However, keeping an adult wood mouse (field mouse) in captivity can be altogether a different proposition to keeping one that has been hand-reared, as it will almost invariably try to escape from its cage. That's not an ideal situation for you if you have two mouse-hunting cats around, who will more than likely continuously seek out every opportunity to get at the mouse in whatever room or outbuilding you might consider keeping it.
My advice is therefore for you to seek out the assistance of a vet and/or a local wildlife rehabilitator to get a professional and knowledgable opinion on this particular individual's prognosis. Whilst I personally would not advocate euthanasia in a case such as this, you need to be aware that is borne out of a knowledge of the particular challenges that face anyone prepared to provide an adequate and appropriate environment for this mouse to live out its lifespan ... not everyone is equipped or indeed prepared to put in the effort needed. Often euthanasia will be seen as a 'convenient' and 'cost-effective' solution to avoid possible or perceived suffering.
Incidentally, the mouse with the missing eye may well have been a better candidate for rehabilitation than this one missing an 'arm'. Loss of stereo vision is probably slightly less of a disability for this species.
Oh yes ... and the situation under the current legislation (Animal Welfare Act) now makes you as the 'keeper' of this wild animal, as equally responsible for its welfare as if it were a dog or a cat. You cannot legally release it back into the wild if it is demonstrably ill, or if it has an injury which renders it unable to cope with independent living. You also assume responsibility for providing it with appropriate care, including relevant veterinary intervention.
By the way ... a shoebox is NOT an adequate container for a mouse. You need to get it into a more secure container as soon as possible ... that is one that it can't chew its way out of, and one that your cats can't get into.
Feel free to PM me for further advice, if you prefer that to posting questions here.