Good question, with lots of twists and turns to keep people entertained. Some recommendations:
1. Carefully inventory and photograph the items. Also, document any information you have regarding them- this includes official documentation of his service history, but don't forget that you are a primary resource to relay stories he told you. Write as much down as you can recall and document from the paper you have, print it, date it, and sign it. This serves as the administrative documentation of the collection and will be useful later (I'll mention more below).
2. Expose the collection as widely as possible. Posting here is a good idea, but also try at places like
www.usmilitariaforum.com and other Facebook and public media groups. As you have already encountered, public museums are chronically short of money, space, and proper receptacles of archives like you have, even if they specialize in a related specific subject matter. But the good thing is that there are many, many museums with rotating displays. The dawn of the atomic age was no small event, and there were thousands of personnel and institutions which participated in the Manhattan Project. At some point, some place connected to this history will want a display. You just need to find that entity.
3. The next piece of advice is going to sound harsh, but be very protective of these artifacts and highly critical of any group which approaches you to handle them. What that means is, don't let them out of your sight or into the possession of another person/entity without
solidsubstantiation that the entity will care for the items.
Do not place them on display anywhere publicly without a loan agreement. This can be simple and complicated at the same time. First, the agreement should be renewable each year, no matter the interval specified for display. Make sure that you keep careful track of the operations of the museum; for example, it is a good idea to join as a member (or stipulate that as part of the loan agreement) so you receive official communications from the museum. No matter if it is a local, municipal, state, federal or private museum, there is a very high turnover rate of personnel in these organizations. The person with whom you sign a loan agreement may not be there 8 months from when it is signed. Then, you really need to update the loan agreement to reflect the new responsible personnel within the organization who are tasked with executing the loan agreement. I can supply you with advice on loan agreements, but be prepared for an earful of anecdotal stories about how they can go wrong. The inventory and photographs are an integral part of any loan agreement.
4. Donation as an option. First thing to know about this is that you are responsible for appraisal of the items in order to receive tax considerations for donating them. The issue with donations is that in most cases, museums require unconditional donations. This means they can do what they want with the item to include de-accessioning the item (selling it) and you have no control over it. Most museums will not accept conditional donations- it is too hard for them to deal with later consequences of dis-satisfied customers, so they generally avoid them. Once again, the inventory and photographs are important because they are part of a proper appraisal.
5. Private sales. While there will be a hue and cry from people who say such things are priceless, there is plenty of evidence on ebay and real auction houses in the US and abroad to disprove them. Privately selling your items to a collector who values the items can and should be a consideration.
Hope this helps!