Bread – way to make this bread recipe’s crust less crunchy

Damn, that looks good. I've never made this particular type of bread before, but here are some things that I do know which might help:

Butter is (roughly) 10% water and 90% fat, while lard is 100% fat. So if you substitute one for the other, you should adjust the amount of water in the recipe accordingly. The flavor will be a little different, but I bet either one would work.

I would not assume AP flour. In fact, I'd assume high-gluten bread flour. The gluten is what creates those fibers. It's the fat that keeps it tender.

This looks very similar to a brioche, so you might want to look at brioche recipes to learn some of the principles. One thing that most of them will tell you is to to knead the dough to develop the gluten fibers, and then add the butter. This I think is the key. I know that fat can keep gluten molecules from sticking together into strands, so if you add the fat later, you should wind up with strands of gluten surrounded by fat.

While it might seem like a drastic step to double the amount of egg in your recipe, in fact, you are not adding that much more egg relative to the amounts of your other ingredients.

A whole large egg weighs about 50 grams and is roughly 75% water so that extra half egg will contribute just under 20 grams of additional liquid to your recipe - you will need to compensate for that additional liquid either by reducing your water contribution accordingly or by adding a little extra flour (like 30-ish grams).

That means you are only adding an additional 5 grams of protein and fat from the egg by including that extra half. You will probably notice a slightly more tender or soft texture in your finished loaf from the slight increase in egg.

Since texture and flavor are so subjective and difficult to describe, the best way I can think of to communicate the possible change is to ask you to think about Challah bread. Challah bread owes much of its characteristic texture, color, and flavor to the inclusion of a lot of egg in the dough. While adding that extra half egg won't bring your recipe into Challah territory, it will advance your bread's texture, color, and flavor along the spectrum in the Challah direction.