Broiling would work, but if the bagel is cold and untoasted, you might want hotter all around. I recommend baking/top-rack/at 500 to get both effects. Given the vagaries of oven timing, I'd start with 5 minutes, checking at 3, just to be safe. Experimental, but tweak-able: If your cheese is not browning, go hotter. If your bagel is burning (but all else is well), lower the temp a bit, open the oven door, or switch to a straight broil.
P.S. Toaster/ovens are perfect for this, and probably worth the cost of admission for melting cheese alone.
You can try to bake both at the lower temperature, and it might turn out OK, but there are a lot of variables that could cause it not to.
I went into some of the effects of time and temperature in this answer. In a nutshell, you have two major processes happening when you bake dough; the first is the Maillard reaction (browning) and the second is water evaporation (drying/hardening). Just to make things even more fun, the second has a strong effect on the first; the more water evaporates, the quicker the browning will occur.
There's also the gluten development (chewiness) of the crust, but that is less of an issue with pizza because you're usually aiming for a crispy crust.
What this means to you is that if the two different pizzas have crusts with different amounts of water, then the one with lower water content may end up rock-hard and even burnt by the time the second one is properly cooked.
350° F is only a little bit higher than Maillard temperature, so the browning will happen pretty slowly; the fact that the box suggests this temperature likely means a relatively high water content, so that the crust can firm up before starting to brown. On the other hand, 425° F and 20-minute cooking time probably indicates a higher water content; you want to brown the crust more quickly before the crust gets too hard and/or starts to burn.
So there's a non-trivial likelihood that after 33 minutes at 350° F, the second pizza (which gives a higher temperature) will be undercooked and doughy. Undercooked is easy to fix in theory, but the problem is that you've thrown off the water ratio, which was specifically formulated for a higher temperature, so it's going to be hard for you to get it exactly right, even if you raise the temperature to finish it off. It'll still be edible, of course; it's really just a matter of how picky you are (the answer to which is probably "not very" if you're eating frozen pizza...).
On the other hand, there's also a not-too-distant possibility that the two pizza crusts are practically the same, and the different brands just have different ideas of how crisp/brown the crust ought to be when cooked. It's really hard to know until you try.
One other thing to mention is that if you plan to bake both pizzas at the same time then you will probably need to raise the temperature/time slightly to compensate. Also, I really wouldn't recommend putting one on a lower rack, because it blocks the heat from the element and tends to mess up what little circulation there is in a conventional oven, and you'll end up with a very flat and probably undercooked pizza.
Bakeries and pizzerias will usually have pizza stones and special ovens with fans and possibly turntables to mitigate this issue; home ovens don't. So I'd recommend either baking them on the same rack (if you have room) or baking them separately.
Best Answer
It's important to distinguish between the two different types of "crisping" that both happen in bread.
The first is the Maillard reaction which is caused by the sugars reacting with proteins; this is facilitated by high heat and low moisture, and is what actually causes the bread to turn brown (and eventually to burn).
The other is simply the evaporation of water - drying it out - which can make the bread or crust noticeably "crisper" without any browning.
The Maillard reaction happens at 154° C / 310° F, which is much higher than the boiling point of water (100° C / 212° F), so the evaporation happens first. If you put a piece of bread in the oven at a low temperature and leave it there for half an hour, it will crisp up significantly but not brown.
So essentially it depends on what kind of "crispiness" you want. High heat will cause the Maillard reaction to occur and that will crisp it up faster, but you have to shorten the cooking time, otherwise it will burn. Lower heat, on the other hand, will crisp much slower - and if the heat is too low, you won't get any browning - but you can leave it in there for much longer and the crust will keep getting drier (i.e. crispier) due to the water evaporation.
The instructions are correct. High heat causes more "crispiness" in some applications, where almost all of the crisping comes from the Maillard reaction or caramelization of some kind, but bread is an exception because of its porousness and high water content (easy for water to evaporate).