Hi Steve,
These days contractor availability is limited, and labor costs and materials are sky-high. One quick and dirty way to get this done is to use clear gasketing. With such a big gap you'd want one on the door and one on the stationary piece of glass. We used this technique on some glass front cases here, in the attached image we closed a 1/2"-3/8" gap. It really cut down on the intrusion of dust though not 100% perfect. We only need to dust them once a year or every other year depending on the case, as opposed to every month like before. I think we got ours from CR Lawrence. I think you'll look for the word gasket or bulb, see how far those search terms get you.
It should be enough to solve your issue in the short term. (read
short term as "good enough so you can move onto the next fire and revisit in five years"). Here is a pic of ours in action. You're looking at two pieces, each with 1 sticky backside. One connected to the front glass, and one connected to the side glass portion.
Consider running a mini-test before going all in. The correct gasket will depend on the angles things meet and the degree of motion. Some come with a flat top, some have a curved top. You'll want to make sure that as the pieces slide on each other they are not dislodging the sticky side, if that makes sense.
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Matthew Isble
Exhibit Designer & Founder of MuseumTrade.org
misble@crockerartmuseum.orgCrocker Art Museum
Sacramento CA
misble@crockerartmuseum.org------------------------------