When boarding a full plane, how can I guarantee that I can keep my handheld baggage with me?

"My bag has medication in which I may need during the flight" would normally be sufficient to get the cabin crew to select a different bag. You bag is more important than other people's.

If your handheld bag is small enough to go under the seat in front of you, it won't be taken from you. If you want to bring a large roll-a-bord to save checking a bag, you can do that, but keep the things you genuinely need during the flight in a separate smaller bag. Also, do not choose a seat, such as the first seat in economy, that doesn't have a "seat in front of you" to use. (Chances are, a small bag would be squished into the full bin for you, especially if it contained medication, but you don't want to take that chance.) With luck, the large bag will get an overhead bin rather than being gate checked, but you won't be relying on luck to keep the small bag with you.

You could wear a fanny pack (a.k.a. hip pack) and keep your meds in that. Then you wouldn't even need to put it under the seat in front of you - it'd just be around your waist the whole time.

Image:

First, make sure the bags you have fall within the airline's official limits. They vary by airline and a few times by aircraft-type within an airline. They always ask to store the biggest bags first, so your chances of being asked will be much smaller if you are among those that simply fit.

Take a carry on and a personal item which is something allowed by the majority of airlines. A personal item has a smaller size-limit and is therefore even less likely to be asked to be checked. In that bag, you put your most essential things, including prescription medications and documents.

Book a seat behind another. This guarantees you will have room for small to mid-size bag. Then, if asked, just tell them you will place it under the seat. Other people's bag do not end up in there, so it would be hard for them to argue. In the end, if you must to, the important line works best for medical reasons, so if it comes to that, just say it nicely. They are usually accommodating.

In addition to very good other answers:

Do not use a rollaboard. Those are the most popular for going to the hold. Some airlines guarantee the first X pieces in cabin then systematically start checking in other hand luggage. Yes, I am looking at you, RyanAir. I'm always one of the last passengers to board and always travel with a backpack. They never ask me to check it in while all rollaboards around me get checked in. In fact, I'm very grateful to RyanAir that they started using this system. I used to have to squeeze my backpack under the seat in front of me (and often that really requires squeezing), while with the present system there is always plenty of place left in the overhead compartments, even when boarding dead last.

My assumption is that they also dislike putting bags in the hold that look like they won't survive some tossing around.

This is one of the biggest worries I have when travelling, and it's undoubtedly because of a less critical issue than yours. I often travel with a lot of (fairly valuable) photography equipment and so I do my utmost to keep it with me at all times. Only had to check them in once so far, and it was because of an unrelated issue. I have no doubt that if you tell them about your medication they wouldn't insist on taking the bag away from you.

If you need more reassurance, perhaps you may want to consider a travel vest. I use a travel vest by Scottevest and can fit a lot of fairly bulky stuff in it alone. I'm not sure about how bulky insulin medication is, but having had a quick look on the internet about some options out there, they would have no problem fitting in it. As the vest is something you wear, it's not something you're going to be asked to check in.

If you let the attendants know that you are keeping insulin that is needed for a medical condition they have to actually do their absolute best to help you with that. They don't want any lawsuits or attention being brought to them about poor customer handling. If they are close to asking it to be put on below the plane maybe ask if they can store the insulin somewhere cool for you on the upper part of the cabin?

If it's really important many airlines have affiliate credit cards that grant you early boarding and thus make it very likely you'll have space. These cards almost always carry hefty annual fees but usually waive them in the first year.

"X airlines is inviting our business class customers, frequent flyers, and those who require additional assistance to board now. Other customers will be boarding shortly."

Have you tried boarding early?