The algae eater is the one you need to worry about. Do not get a Chinese Algae Eater. As they get older, they grow more aggressive and will try to suck the slime coat off your beta's fins. It could kill your beta.
Betta and algae eaters, such as plecostomus, usually do fine in a tank together. The algae eater will not bother the bettas fins, and the betta is unlikely to harass the algae eater, and they are very hardy fish.
Betas coexist with other fishes which donot fight with them or harm them. So there is nothing to worry to put a beta and a algae eater in a fish bowl together. But both fishes should be kept in a decent fish tank with a filter. But remember not to put two male betas in a fish tank.
It depends on the nature of your beta. The males are definitely aggressive towards each other and they can go either way with other fish. Just make sure to observe them for a while to ensure the safety of your algae eater.
If the fish bowl is big enough, it may be okay. However, beta fish are aggressive, especially the male betas, so keep a close eye on them for a few days. Luckily, betas are small and probably could not hurt the algae eater.