Most of these type of shelters keep a few spaces open for emergency cases. These are the things like owners dying/being forced homeless unexpectedly(like a fire or tornado or other disaster). If the shelter you want to surrender to emergency space available then it may be possible to do what you have suggested, even though if asked outright most of these shelters would deny that such accommodations are possible.
Some of the things that will influence whether or not your pet is a candidate will depend on the age and breed of the animal you are surrendering. Younger and smaller animals are much easier to adopt out than older and larger animals. Also breeds that have a bad reputation are unlikely candidates. That said pretty much everything has a price. So if your pockets are deep enough you may be able to get a discriminating shelter to take a viscous animal of a notorious breed.
That said chances are it would probably be cheaper to board the animal with a quality boarding facility. For such an accommodation to make sense it would need to be generous enough to cover the cost of taking care of the animal you are surrendering indefinitely, or at least for far longer than your animal is likely to need to get adopted. So a cute little puppy of a small breed is going to be much cheaper to buy in than an older pit bull mix.
The shelter is highly unlikely to give you a number, you are going to have to make an offer that is seen as quite generous. While a large donation can be stretched quite far there can be side effects when inevitable the trustees and other large donors find out about the accommodation. Most shelters run primarily on the donations of just a few or even just one large donor. The other donations help certainly but and are also critical to keeping the shelter going but the reality is the heavy lifting is done quietly by a few, usually anonymous, individuals. Your donation will be large enough that it will not upset these large donors.
How should I go about making the suggest for best impact and how much
should I offer?
I think your shelter does great work and if you are able to make a one time accommodation for my surrender I would be prepared to offer a donation of $X to help defer the costs and offset the problems caused by your making the exception.
You will get one shot at this offer and if it is not enough any increases are likely to be seen as an insult. The first offer is an offer to provide great assistance the second increased offer is a bribe to someone you do not truly respect. If you are willing to go higher then go higher the first offer. If the offer is rejected accept this rejection humbly. Remember you are asking a huge favor not attempting to purchase a luxury item.