Have you tried ski goggles? Good ones aren't cheap but may work better. I haven't found any utility goggles in hardware stores that worked well for bicycling.

Look for lenses that meet safety standards agreed upon by ANSI, ISO, ISEA. Usually those will be tough enough to resist scratching and scuffing longer. Or look for safety glass rather than polycarbonate.

There are also some retro style cycling and aviator goggles, the type with separate lenses for each eye for that classic bug-eye look. Might be easier to fit a mirror. Good ones will be pricey.

I mount my Take-A-Look mirrors on my helmets, since I switch glasses often. There's a little trick to using the Take-A-Look on a helmet:

• Cut out a stiff plastic card to substitute for the eyeglass earpieces. Those junk membership cards from grocery stores work well.
• Cut it to fit a space inside the helmet, usually between the EPS foam and MIPS support or liner.
• Zip tie the mirror to the trimmed card.
• Zip tie the entire rig to the helmet itself, through the vents.

This will leave enough wiggle room to adjust the mirror stalk to suit my riding positions on my upright hybrid and road bikes, with or without a bandanna on my head (usually with). However I can't wear a cycling cap under my helmet -- no matter how I adjust it the bill either blocks my mirror, or forces the helmet so far back on my skull that I can't see the mirror. I don't care for cycling caps anyway.

There are other mirrors better suited to helmets, with articulating support arms. But I like the smaller Take-A-Look and this mounting method works well for my helmets. Most other helmet mirrors are too large and would block too much of my view.