I'm a plumber, you're fairly spot on but 5 things to consider.

1. If a plunger didn't work then draino is unlikely to work as the clog is likely past the trap and vent meaning that the draino will be very diluted if it ever makes its way to the clog. If it's just a slow drain the draino will drain past it and not help.

2. Draino can weaken pipes. They swear up and down on the bottle that it doesn't, but it does.

3. People often have leaks under their sink that they didn't know about. Now there is draino covering your cabinet and floor, possibly leaking into the ceiling space of the floor below.

4. Draino has lye in it, your pipes have fat and grease in them. What do you get when you mix them together? Hard soap. Do it enough and you've made you problem a lot worse, it can also strip the oils from hair and leave behind the protein eventually causing a bigger problem.

5. It weakens auger cables. Weak auger cables can break off in pipes. Your problem just got a lot worse.

I started with 2 things to consider but more just kept coming to me. Seriously, try plunging, try using a hand held auger, don't pour boiling water into any of your fixtures, baking soda and vinegar won't do anything (but if you insist on trying DO NOT MIX WITH DRAINO)

What kind of water is it pumping? Sump pumps are dead easy. If you undo that union you can pull the whole pump out. Buy a new pump with the same dimensions, spin the MIP out of the pump, Teflon tape it, spin it into the new pump and put it back in. You might also find out that the pump is just jammed or only need to replace the float.

Edit I see it's pumping rain water into the house, cool what for?

Anyways I'm surprised you can't get a plumber this sort of thing is easy money, you should be able to figure it out yourself.

See if you can find dandelion honey. Boil some dandelion flowers and sweeten it with the dandelion honey. Probably going to be the closest non-alcoholic facsimile.