Can you try Machine Embroidery button hole pattern? Also try a teflon sheet (makes the fabric slide easily, used for free motion quilting). Perhaps frogs? (The asian inspired knotted closures) or really large hook and eyes with a button on the outside for looks? Lots of choices, some require more planning like having the teflon sheet, but you should be able to come up with an alternative. Good luck! Linda

How good is your hand stitching? You can look up how to do a button hole stitch by hand. Since the coat is "old fashioned" it might work. Use a thick contrast thread if (you have confidence in your stitching).

I was going to suggest handmade buttonholes too.
You could undo the seam at the front edge and insert some sturdy cording and do loops for the buttons. If the fabric is really too thick, the buttons might pop out of buttonholes anyway.

have you tried lessening the pressure on your presser foot? This will help thick fabric move thru easier.

How about toggles?
AlmaG

Why don't you just use braid closures on the coat? Joann's has them, see link below.

How about old fashioned bound buttonholes? If you don't know how to make them maybe you could find a local tailor or seamstress that you could trade some embroidery work for buttonholes???
Another alternative might be to use heavy duty sew on snaps on the inside of the coat and just stitch the coat buttons on the outside so it will look like it is buttoned.