I have a lot of experience with this, ranging from personal to family members with huge amounts of allergies and asthma (ranging from severe to mild). The conversation starts with how bad is the asthma? Is this more of a turns blue pretty often, or just some mild wheezing when sick, or perhaps wheeezing after a lot of athletic activity? If severe asthama, such as taking an inhaler puff multiple times a day to be able to live, then you might be in a bad spot, but you would have to test to find out (find a local GR breeder to see if you can hang out). If it is this bad, accept that there is a chance that if you try to get the dog you might have to find the dog a new home if you had too (which is normally not too hard with a GR). But ask if there is someone willing to do this job of being the bad one to say the dog has to go.
But if the asthma is not that bad, there is hope. How are your kids allergies? If they are around lots of allergies, are the allergies more itchy eyes, runny nose, sneezing, or are they throat closing asthma inducing. Is they are less asthma inducing and more sneezy, there is hope.
If you decide to try, which to me there is a chance to try as long as the asthma is nor severe, then you will have a few things to do. You will need to vacuum more often, regularly brush your dog (cant stress enough), and dust more throughout the house. Dont allow dogs on furniture or bedsalso. If you have carpets in your house, that is not as great as a solid floor, but still can be OK.
With GR, as with lots of dogs, there is a good chance that asomeone in your home will need to go on full time allergy meds that they dont currently take, which is not a big deal. A chance that someone might need to get allergy shots, doable, but not as great.
Lots of rambling. But yoru situation is maybe not as bad as you worrry about. Most people, including your kids, will sacrifice taking a pill to have a dog