Options that work:

You can buy chlorine tablets at a swimming pool supply. Same thing as bleach just takes the tablets a little longer to dissolve.

You can buy copper sulfate based products at pond and swimming pool supplies. Copper lasts a long time. A copper molecule will attack an algae cell and then be released to attack another. Where as a chlorine molecule attacks a cell and both are destroyed. So copper lasts longer. If you're concerned about poisoning birds you may want to buy a copper test to make sure you're not adding too much. It's not considered to be super toxic to birds and animals, but not good either.

You can place a 1 GPH drip emitter that runs 24/7 into the pond and let the pond overflow. An overfloat (pretty simple pipe) can be installed to move overflow away from the pond to water loving plants (any marginal pond plant) or even trees, shrubs whatever. This would be safe for plants, birds, people, pets and may even be safe for fish although 1 GPH for 200 is pushing the limit for fish if you have chlorinated water. The cost in Phoenix for 30 days of 24/7 1 GPH drip is about $5 while in Rochester NY it's about $2.50, San Jose CA about $3-4. Nice thing about 24/7 drip is once you set it up you have nothing more to do. Don't even have to ever top off the pond. About as easy as it gets.

You can turn the water feature into a pondless feature. Place the pump inside a bucket with holes and fill the pond with rocks/pebbles/pea gravel, whatever look you want. Direct the pump output to a fountain of some sort. I always add an auto fill just to make life easier.

You can add a UV filter, about 8 watts so not much $$$ to run. Cost about $100. Bulb has to be replaced yearly (about). You do have to get the flow right so I always install a bypass so flow can thru the filter can be controlled. UV is 100% effective when installed and maintained correctly.

Things that don't work:

There are a ton of products that claim to be magic...they aren't. Some will even make a real mess. Bacteria in a bottle, anything with the word barley, you name it.

There are a ton of magic DIY things. Magnets, caterpillar droppings, plants starving algae, etc. Ponds so often clear on there own and when they do the owners sometimes dream up reasons and fully believe this crap and tell everyone they can. Caterpillar droppings is one of my current favs.

You don't mention the kind of filter you're using. Unless it's a UV you don't need it. Filters, like bacteria in a bottle is another fav scam retailers used.

Pump doesn't have to run 24/7. Pumps have no effect on algae. It would be needed for a UV filter.

The slimy green stuff can be lots of things. Probably dead suspended algae that was turning the water green. There's no reason to scrub it unless you prefer the look of your liner. There are other methods for removal if you like. In general this type of water feature would only be cleaned once a year by draining the feature and shop vacuuming out the crud or scooping. The crud doesn't really contribute to green water.

Thanks for all the information. Think I'll give the copper sulfate a try.

I'd prefer to keep the pump/filter going 24/7. Don't want the water to be stagnant and attract Mosquitos. It makes sense that the filter really won't clean the water. It's more to protect the pump - keep large particles out of the pump that may damage it.

It's a myth that a pump moves enough water to deter mosquitoes to any meaningful degree. The copper sulfate will kill all mosquito larvae. So really you want mosquitoes to be attracted and laying their eggs there instead of some place where the larvae can survive.

I don't know the exact set up you have for the filter, but if you tire of cleaning the filter you can likely just set the pump inside the pond on top of some bricks or whatever. Keeping it off the bottom is enough to stop most clogs. I used to use prefilters to protect the pump but really they clogged more than the pump did without the pre-filter. So I stopped using per-filters and it made life much easier.