Anytime Tools 5 Piece 3/8" MINI LATHE INDEXABLE CARBIDE INSERT TOOL BIT SET
You could take the blue layout dye and de-burring tools off your list and that would almost pay for these carbide tool bit set.
I use the cheapest blue markers I can find at Walmart for layout dye.
And you can probably find something laying around to deburr with. In fact a small set of hobby files would probably come in more handy.
You might think about going around to yard sales too. You can probably pick up things like files pretty chaep at yard sales.
The carbide tool bits are a far better place to put your money. I don't think that lathe comes with any tool bits. It might come with some High Speed Steel tools which are good to have too. But then you'll need a grinder to sharpen the hight speed steel tooling. Don't worry about buying a grinder. Again, that's another item you might be able to pick up at a yard sale. You don't need a real good grinder. The cheapest one you can find will probably work.
You'll also want a drill chuck for the tailstock.
MINI METAL LATHE TAIL STOCK DRILL CHUCK TOOL WITH MT2 MORSE TAPER ARBOR MT 2
You'll definitely want the tailstock chuck so you can drill holes through the center of rods, etc.
That's such an important item I'm surprised it didn't come with the lathe. But at least the lathe does come with a center in the tailstock.
That should get you started pretty good.
It wouldn't hurt to have a hacksaw and a vice. Again, yard sale is your best bet for those items. You won't need a large vice. The parts you are making are small. Any clamp-on hobby vice would do. You should be able to find something like that at a yard sale really cheap. When you spend the money get things you aren't likely to find at yard sales. Like the tailstock chuck and carbide tool bits.
Don't spend money on anything you can make do with used or otherwise. Like I say, a simple marker pen will replace the $13 layout dye. It doesn't even need to be a blue one. Any color will do.
If you're going to be pinching pennies, pinch them in the right place.