I do not think the tank is able to run non CO2 methods.
Too many fish.
The "trick" with a good well run non CO2 tank is balancing the fish load to less than the plant demand or slightly less etc, then you can over feed or dose a tiny amount here and there.
You might not even "need" to do that, with good reason, need is rather subjective and the amount of nutrient stress at low light and under Non CO2 growth rates is very small, so issues will occur very slowly and perhaps not much at all, CO2 is more limiting generally.
As far as the fish load, you are not willing to remove the fish, then you need to one of two things here.
Add a bit more light, a pair of T5's would do great.
Add CO2 or go with dosing Excel,CO2 gas tanks will do FAR better over time for you and cost less. you can do the Excel in the meantime till you buy the lights and the CO2 gas tank.
Get the gas BEFORE you get the lights, other wise you will have lots of algae.
Now you can do water changes all the time and can use something like EI dosing etc. The dosing is simple then and you can tweak to suit.
In non CO2 methods, one of the main tenants is NOT doing water changes to maintain CO2 stability, plants are well adapted to low CO2, adding it weekly etc will cause the plants up and down regulate their CO2 uptake mechanisms slowing growth whereas algae will just bloom as CO2 acts a good signal to induce spores to germinate.
So you have to balance the fish waste and plant demand with this method. This is a balanced
You are not going to have any balance with overloaded tanks, the only practical way there is to do water changes/dosing etc.
That's about the only way to keep it stable with high fish loading.
Why not be kind to the fish and give them a proper home and not over load the tank?
That's the real solution.... one that reduces the errors we make and prevents fish from dying due to our neglect and not remembering to care for the tank every now and then, like everyone of us at some point does.
So fix that, then you have far more choices of the method you wish to do and also a lot less work and cost involved.
Put this way, you could have 2 tanks and go non CO2 vs the one tank that's CO2/weekly water changes etc based on cost. This way you could keep the fish and have more planted tanks and less work.