E.G.
First of all, I'm sorry that Lucy is going through this, as is the whole family.
O.K. now here is my background: I am a licensed Veterinary Technician with about 20 years experience working in Emergency, ICU and anesthesia. I have seen very many dogs with the condition that Lucy has and actually euthanized my first girl, Taro, at age 13 due to splenic hemangiosarcoma. I have professional and personal experience with this is the point of all that.
Please don't make Lucy get to the point where she ruptures her spleen-she would go fairly fast but it's not peaceful or painless and I'm surprised your veterinarian insinuated it is. If you have the choice to euthanize her before that point, please show her how much you love her and don't let her suffer.
As far as your beagle goes, I do think dogs grieve for each other but each in their own way. Since I am in the profession, I have had the ability to euthanize my dogs at home and then I let the other dogs say goodbye-usually they sniff their brother or sister some stay far away and that's fine, too. I had one bring a toy to their brother. I think they understand death-certainly in different terms than we do but perhaps going back to their wolf ancestors, they understand something. If you can't euthanize at home, your beagle will adjust but will probably need some extra loving and TLC. Also, don't quickly take away all Lucy's stuff (bed, bowl, leash) but maybe take one away at a time, I think that might help-though might be harder on you. I also think your beagle might really help you grieve so cry and hug him, use him as a tissue! But just make sure Lucy has tons of loving and not waling cries in her last days and then just love and attention to your beagle. And your human child will just be worried about your sadness but probably not grieve Lucy but he will be a great source of love for you and your husband plus he will keep you "normal" and on your schedule, reminding you to keep going on.
Best wishes and peace to Lucy and the family,
E.