Well it's perfectly possible, but the fact of the matter it's fairly useless. Aside from it being kind of cool, no one with the knowledge of how to do it would want to waste multiple hours finding out how to properly flash the BIOS of a GBA.
This would undergo a few steps. First you would have to find out exactly how to flash it. Then once you figured that out, you would have to pretty much go through the code already on there to determine what does what. Then after that, you have to custom write your own bootloader to flash onto the BIOS, with your custom animation, and if you do it wrong, it's likely you wouldn't be able to play any game. So essentially it's possible, but no one will do it.
Your best/easiest bet would be to just develop your custom cartridge with your custom boot animation on the cartridge. The generic Gameboy Advance boot animation would show, then after that would be the boot animation for the game, which you could make custom. But even doing this much would be a pain.
Any reason you want this, other than just wanting a cool custom boot animation?