My comment goes to the BSA organization broadly.  Individual COs and the COR continue to be able to assure unit operations are expressive of the particular faith values of the CO.  

I like @scotteg83's suggestion if the strips are still available. If they aren't ... I met one troop who had really sharp embroidered name name tags. They sacrificed one shirt for a seamstress to use for material to make the tags. This made the background an exact match to the shirts! Doing something similar for these letters sounds like a good idea. You could talk to your seamstress if it's worth the trouble to try and lift as many letters as you can salvage so that she can sew/glue them on the material from your sacrificed shirt, or if it's just as well to embroider new patches or just sew them onto the shirts directly. There are probably other modern solutions like digital-to-cloth printing/transfer, but I'm not sure they would be cost-effective.

What type of organization is the CO of the new unit?  Isn't it the CO who decides what "values" the new unit follows (in addition to the values of Scouting, of course.)

I have not had much contact with LDS Scouting as a program in my 30+ years as a Scouter, as it had often been conducted in a somewhat separated manner -- such as LDS-only weeks at camp when non-LDS units were not present.  I have had many pleasant interactions with LDS adult leaders in activities above the unit level.  I have nothing but gratefulness for the many years of LDS support for Scouting and wish them the very finest as they go a different way to serve their own membership.  For all sorts of reasons, they have established a firm cut-off, and that is appropriate from a legal and program standpoint.  Businesslike action and language used sever Scouting from their church is unambiguous and fine with me. Those of us who are religious and belong to other faiths know the unstated reasons why they are departing us, because LDS Scouters with informed leadership roles in their church have been expressly telling us for many years.  The contest of wills regarding whether the BSA should embody the views of particular faiths (not just LDS) is now over. No longer will Scouters be discomforted by seeing those arguments play out within an organization that respects and requires faith among its members -- but is neither a church nor the arbitrator of particular beliefs. For those LDS folks who remain with us, you have our continued and fast friendship.  You will love our program just as much as you always have -- and maybe a bit more.  For anyone still trying to conform the BSA generally to a particular religious, political or "cultural war" view, you will be unsuccessful, disappointed and should probably reevaluate your personal rationale for continued membership.  We have decided to welcome everyone.

our pack has done it several different ways the past few years. The Webelos 1 parents are responsible for B&G every year with anyone else that wants to volunteer.  The year before us got a caterer and it was nice so we did the exact same thing.  The food was dropped off to the church basement.  Chicken, spaghetti, rolls etc..  $10/person, scouts were free.   All money covered costs. I don't recall the price but the next year there was a chef in the group so they had a similar meal but he cooked everything in the church kitchen. The year after that it was decided that a big meal was too much work and they kept it simple with a nice sheet cake and beverages.  The pack paid for the cake and there was no parent fee.  Take a pole of the parents.  Give them a few options with estimated family cost and see what they say.  B&G should be about the scouts and a scout is thrifty so I like the cake only idea.  It's easy and fast so it takes less time from the volunteers/planners.