youll need an N95 disposable or use a respirator
https://www.amazon.com/3M-Facepiece-Respirator-6291-Particulate/dp/B000FTEDMM/ref=sr_1_10?
Since you're trying to work a tight budget, I would skip the Sorby tools and opt for some less expensive HSS options. Sorby makes some of the best HSS tools on the market, but the price premium is rather high for the name. Personally, I'd suggest you opt for Hurricane as starting tools and then make the decision to go for more expensive tools down the road.
Hurricane are solid inexpensive tools and you can pick up way more tooling for $14 more than the tools you have listed. That set gets you a bowl gouge and parting tool, which are both fairly important. Take one of the scrapers and regrind it into a negative rake scraper - I'd use the 1" since a scraper that large is very likely to stall that HF lathe.
I'd also recommend spending the extra dollar now on a respirator and some P100 cartridges. This is the sort of thing you really want to have up front and is far more important than a sharpening jig - you can always learn to hand sharpen tools, you can't un-COPD your lungs later on. Many people who learn freehand sharpening swear by it and prefer it anyway. Luckily, you can pick up a solid 3M mask with filters for under $20 on Amazon.
So, for about $35 more than you're currently looking at, you can add some safety and get way more tooling for your money, at the expense of having to sharpen a little more frequently.
it's this one - the one you linked doesn't filter out asbestos. I bought the same one for my first trip myself, but this one I've linked definitely works
3M provides a HEPA mask which I've found to be great. A dust mask won't work, you need a HEPA.
Safety glasses and a big face shield are critical. If I'm grinding a lot or grinding ANY thoriated TIG electrode I'll add a respirator with the pink P100 filter cartridges. Don't forget hearing protection! If you wear earmuffs then behind the neck ones like these will fit with all the other gear.
My bad, I didn't notice the delivery options. Crazy that Amazon could have stocking issues from how big the wildfire problems are.
You can also try mixing up search terms for disposable/nondisposable masks and higher NIOSH ratings - there's one $20ish for an older 3M half mask model + 3M P100 cartridges, one and two day shipping available if I read it right.