I'm not sure why you're using fungicide on mealy bugs! Wouldn't an insecticide be better?

That said, if your fungicide contains neem oil, then it should still help. Neem oil can be used as an insecticide too. However, because of the way it works, it can take a while for you to see the effects.

If it's still not working after another week, try sponging the whole plant down with a 1:1 mix of water and rubbing alcohol (isopropanol). That kills mealy bugs pretty well.

Thanks - so, just to be clear, I'd make a solution of 1:1 water/rubbing alcohol, apply it to a sponge, and run the sponge over the entire plant?

Yes, exactly!
The entire plant, including stems, under leaves, everything except the soil and roots. Avoid getting alcohol on the roots. I don't know what it might do.

And just to be doubly safe, test the solution on one branch first and leave it over night to see what happens. Very occasionally, plants can be damaged by isopropanol, but it's unusual.
The only time it's ever happened to me was with a sick streptocarpus, but you never know...

I like to use a spray bottle filled with rubbing alcohol and just spray the plant.
Mealybugs can hide in the soil and on the pot, so you really should thoroughly cover EVERYTHING.
I personally have not had any issues with rubbing alcohol harming roots, and I think it's worth the risk in exchange for eradicating the mealies.

In fact, they can even hang out on the furnishings nearby, so you may want to relocate the plant when you start treating it with rubbing alcohol.

Summersunshine, that's good to know!
I've always been worried about isoprop. on roots, but now I won't!

Rubbing alcohol should not be sprayed on the roots. Do not drench the soil. Root mealybugs and foliar mealies are not the same critter. The former aren't a common houseplant problem, thankfully.

I've never seen mealybugs in the surrounding environment away from a plant. Their eggs are located in the sticky white cottony blobs and aren't easily spread from one location to another.....except by us. Most people don't know that the eggs are in that icky stuff.

Sponging or wiping them off is fine, but be sure to rinse the cloth in between swipes. I recommend misting with the isopropyl solution, too. ...I usually mix mine at a ratio of 1:3 (alcohol to water), but have used stronger solutions with no problem. You'll be targeting the small nymphs.

Mealies can find all kinds of places to hide on a bamboo palm, as you've discovered. That's why going on a hunting and removal mission can be helpful....maybe with a little dipping bowl of isopropyl and some q-tips. Plus mist the entire plant, into the nooks and crannies, every few days.

Bamboo palms are prone to the three dreaded pests of houseplants, scale insects, spider mites, and mealybugs. Your neem and the alcohol will help control all three. But don't let your guard down. It's nearly impossible to complefely eliminate these pests but you can gain the upper hand.

Firefly, be careful about combining and using too many mixtures on your plant. You don't want to damage your plant. Your fungicidal product product is also labeled for use on several pests, but don't know about mealies. Neem is the active ingredient and I see no reason to double up with your own neem spray.

Good luck. You've got a challenge ahead of you!