The plants I mentioned wouldn't take over the tank at all. They're considered low light tolerant plants and grow pretty slowly (maybe java moss grows fast at times). Aside from the plants I mentioned, you can also consider dwarf hair grass, marimo moss balls, jungle vallisneria, and dwarf sagittaria. For the DHG, I'd get eleocharis parvula since it's the shorter version. You can research the plants and see what's best for placement as a foreground, midground, and background plant.
For lighting, with your criteria and budget, I'd definitely get the Finnex Fugeray 20" LED fixture.
Amazon.com: AL-20WM: Finnex FugeRay Unibody Ultra Slim LED Fixture - 20" Daylight +Moonlights: Pet Supplies
Here's a couple reviews and people who use the Finnex Fugeray:
Finnex FugeRay Product Review
4g Schuber-Wright shrimp tank
Note: You can see their plant/flora list and also consider those if they're not in my list. The light fixture I recommended is perfect for low to moderate lighting. Both examples I linked to are non-co2 injected tanks.
I recommend putting your lighting on a timer. It makes things so much easier and your plants will get the needed light (photo period) even when you're not home to turn the light on and off.
Also make sure to add about 2 to 3" of substrate to provide enough depth for your plants to develop a good root system. This applies to all the plants I mentioned with the exception of the Anubias, Java Fern, and Moss (of course). Anubias, Java Fern, and Moss can be attached to DW using cotton thread or fishing line. I prefer the cotton thread because it will disintegrate over time, which the plants will have attached themselves with their roots by then.
I do, however, recommend a good liquid fert dosing. You can go with
Seachem Flourish,
Excel, and for Iron and Potash, you can get
API Leaf Zone. Just follow the directions on the bottles for easy dosing. Hope this helps!
So what are the specs on your goldfish tank?