Suction cups do tend to just stop suctioning over time, so a simple solution may be just buying new ones. In my experience with them, I've found that sometimes they'll only stick to the surface if it's dry, especially when I'm dealing with a plastic surface. I suggest you try drying the bucket, and then attempting to stick the suctions then. If that doesn't work you may try to buy new ones, they're pretty cheap, and serve lots of purposes, so if they don't work it won't really be a waste.
Since you only have brine shrimp in there, you can just place the heater on the surface like you've already done, the weight of the heater will be enough to hold it in place provided the filter's output isn't strong and/or directed at it, and of course, the brine shrimp can't agitate it.
Back to suction cups. Suction cups work differently on different types of surfaces. You can try placing a flat rock (which can also serve as a decoration), and then suctioning the filter/heater in place there. An alternative can be placing a heavy (but not too heavy) sinking object (anything really) over or next to the filter or heater. This will serve to keep them in place and may also be used as a decoration, or a way of concealing the filter/heater, and making the setup look better(depending on what you're using to keep them in place). The only real disadvantage here is the space consumption.
If you're not planning to to use this container for anything else, you may as well go to more permanent measures. You can drill holes in the wall, then screw a stand or support into it, and secure all that with lock nuts. Top it off with siliconing the screws completely from both sides to stop any potential leaks and the metal from reacting with the water.
Other than that you can look at any equipment you previously had and isn't functional or needed anymore. I can't provide you with any ideas here since I don't quite know what you have, but look for previous equipment which serve similar or complementary purposes: parts from them will probably be the most helpful. Of course, through the whole process make sure everything is aquarium safe. As for packing tape, it's not really toxic, but it's still not water resistant and will loosen quickly, so it's not a viable option.