You certainly don't sound overstocked, but knowing your water parameters is always useful. The cause of your algae is more likely to be down to light.
Make sure your aquarium is as far away from natural light as you can reasonably get it. Far side of the room, 3 to 4m away from a window. Or keep the blinds/ curtains closed. And reduce the time that the aquarium lights are on for. 6 hours is usually sufficient for healthy plant growth, if you have live plants. I have an aquarium with the light on for 4 hours per day and the plants do fine. If you don't have live plants then reduce the light even further. Just have it on an hour or 2 in the evening when you are home.
If your filter is cutting out there is something wrong with it. How long have you had it? What is a regular maintenance schedule and what maintenance do you do?
The problem with getting a bigger tank due to livebearers breeding is that it doesn't deal with the issue. It just kicks the can down the road. Guppies breed prolifically, you start with a few, they make babies, then the babies start breeding with each other and their parents and make more babies, then those babies start breeding, etc etc etc. In a short space of time your aquarium is overpopulated and then you move them to a bigger aquarium, and then that one gets over populated and you need more aquariums or an even bigger one.
The only way to stop this happening is to separate sexes. And as soon as the fry are old enough to sex, separate those into M/F. And females store sperm so can keep producing babies for a year. So as new fry show appear keep separating them. After a year or so you should stop seeing fry if you keep males separate from females.
We have another member who had a few guppies in small aquariums, they bred and they too got bigger aquariums rather than deal with the issue. Their plans to sell the fish didn't work out, they can't even give them away. The hobby is full of people trying to get rid of guppy babies and not enough people wanting to take them. And the fish just keep growing in number and various plans to deal with the issue never really work out.
Gotta deal with the problem at source rather than just trying to accommodate an ever growing population of fish.
Can you post a photo of these alien fish? Someone might be able to identify them for you.