If you bought the plec from the shop on the basis that it would clean the tank then that was a mistake. Plecs don't clean. They eat veggies and poop for England!
joking aside, its not your fishes job to clean the tank, its yours. Also, its likely you have a common plec in which case, even with your large tank, it might be wise to take it back - they get 2 feet long. And poop accordingly (imagine 4 foot long strings off poop across your tank)
The other option is a serious filter upgrade to ensure it can handle the bioload of the plec. I would suggest for a tank that size an external filter rather than an internal one. You can buy them second hand off eBay, or all pond solutions and hydor do good external for relatively little money. A small internal really isn't sufficient for a big tank like yours. Also, for the record, healthy adult fish don't get sucked into filters, so I would suggest your neon tetras were weak rather than the filter being overly strong.
Next, how are you cleaning your gravel? When you do a gravel vac you need to push the end of the vac right down into the gravel and you'll get a cloud of muck come up the tube, once the cloud has moved up a couple of inches, pull the vac out and push it down somewhere else. You should get quite adept at this, and be able to get around all the available gravel with a 50% water change. This will keep on top of all the poop on a week-by-week basis, then as previously said, once a month or so lift the ornaments up and vac underneath them. Maybe even do one ornament each week, so it's not a huge extra job each month.
You are quite right to swish your filter media in old tank water, not tap water. Chlorine is toxic to filter bacteria as well as fish, so the dechlorinator is for two reasons! Keep up with that
I'd like to suggest two other things that will probably make life easier for you. Firstly: put a hose attachment on the end of your gravel vac and use a piece of garden hose to empty the water directly into your flowerbeds/water butt/drain, and then refill directly from the tap. Trust me, it makes water changes SOOOO much easier! (I have a 240l tank and cannot imagine doing water changes any other way anymore!) Secondly, buy some concentrated dechlorinator, like Seachem Prime. It's slightly more expensive at first, but it will last you much longer. I used to use stress coat, and was going through 1 bottle every couple of months. Then I bought my first bottle of prime. That was nearly 2 years ago and I have just bought my second bottle yesterday!
Finally, I just want to reiterate that if you are getting ammonia/nitrite readings the way to fix it is to do water changes, not to add bacteria liquid. Those things are a waste of money. Save your money and spend it on something else. Nothing can substitute for a healthy bacterial colony, so that is what you need to achieve
So, to recap:
1. Return plec if possible.
2. Buy a bigger filter, external if possible.
3. Keep up with your water changes.
4. Gravel vac every water change.
5. Use a hose to make life easier.
6. Use prime to save your self some pennies.
7. Stop using bacteria liquid to save even more pennies!
