You have the theory correct but I am not sure if a few freeze dried shrimp will rot into enough ammonia to start the cycle let alone produce enough ammonia to imitate the daily poop load / bio load of a tank full of fish or even half a tank of fish. Should you be successful in getting ammonia from a few freeze dried shrimp you will only grow a limited number of bacteria. A good start no less. We should be adding roughly the same amount of dry food as we would to feed a tank load of fish each day. It will not rot into ammonia for a few days but each day more and more is added. No more than 4 ppm
of ammonia in an uncycled tank or you may poison the living bacteria.
This will sound silly but never the less. Next time you routinely clean the filter in the 6.5 gallon . Wash the dirty filter into the new filter or as close as possible. The nice new tank will get some real true bacteria from some of this rinsing. Don’t worry about the colour of the water in the new tank the filter will pick up what does not settle onto the gravel.
If you have dirty gravel to spare add a little of that to the new tank. Put it in a ladies stocking/ sock if you don’t want that colour. An old ornament will also transfer some real live bacteria as bacteria lives ON all hard surfaces.
An Australian would call a 6 inch king prawn a shrimp. Some countries call a 1 inch cocktail prawn a shrimp. You need something big and smelly. Again I am not sure if you will get much ammonia from a freeze dried anything. Only a water test will tell you if you have ammonia in the tank . Also cheap to see if you have ammonia or nitrates in your tap water.
To speed up cycling you can run the tank a little warmer than you would for tropical fish . ( say 29 centigrade?) You can also run the water a little low (2 inches) so you get plenty of splash from a HOB
filter. Not miracles but it might help a little.
Now to confuse you a little. Sorry.
You could move all the fish all the little filters or cartridges/ foam/ sponge inserts . Move all the ornaments all the gravel and all hard surfaces to the big tank . The little filter running beside the big new filter. It’s up to you to consider this idea. Ask around but it’s what I would do.