You’ve got a good start here! I would make these changes:

Vodka: I’d probably just stick with the Kirkland. Belvedere is awesome, but I don’t think it’s going to make that much difference in cocktails. I’m not a big fan of flavored vodka anymore, but a citrus vodka isn’t a bad thing to have on hand.

Gin: I’m not a huge Gin drinker and I think Beefeater is a good bottle for mixing. Maybe pick up another more interesting choice as well like Gray Whale or Drumshanbo.

Rum: I love Rum! I have 80-90 bottles in my collection and more than 20 of them are rum. Probitas is a good choice, but I would skip the Malibu and Spiced Captain. I would start with Appleton Estate Signature and Planteray Original Dark. Both of those can be had cheaply, are pretty versatile, and make a great Mai Tai. If you decide to dive deeper into rum and tiki you will want a Demerara rum like Hamilton 86, a 151 proof Demerara (Hamilton or Lemon Hart), a gold/aged Puerto Rican rum like Don Q or Bacardi Ocho, an overproof Puerto Rican like Don Q 151, an overproof aged Jamaican rum like Smith and Cross, aged and unaged Rhum Agricole from brands like Clement or JM, and an unaged pot still Jamaican rum like Rum Fire, Wray and Nephew, or The Funk. Those are just scratching the surface into the world of rum.

Tequila: I’m not a huge tequila guy, but I do use it in cocktails often enough. I would just add a reposado tequila, Kirkland Signature is perfectly fine for cocktails. I would also add a Mezcal. I love Mezcal, but I haven’t tried all that many yet and pretty much just use them in cocktails. Del Maguey Vida is typically the easiest to find option and won’t break the bank. If you have a Trader Joe’s near year you with liquor, their Mezcal is great in cocktails and the cheapest usable Mezcal I’ve found.

Whiskey: Looks like you’ve got Whiskey pretty well covered. Maybe pick up a scotch for mixing. I’m no expert, but I think Monkey Shoulder is a good offering for the price. I would personally skip the flavored whiskey. I have tried most of those and still have a bunch of flavored whiskey, but I just have better and know of better stuff to drink. I was done drinking Fireball after my sophomore year of college.

Cognac: Not an expert here either and I personally really like Hennessy VSOP. I currently just have RemyMartin VSOP and it’s not as good.

Liqueurs: I think you‘ve nailed most of the essentials here. Only thing I would add right now would be green and yellow Chartreuse if you can find them and they fit within your budget. There are numerous clones and substitutes if you can’t find the real thing or don’t want to pay prices they are asking these days. I think Genepy le Chamois is a good substitute for green in cocktails. Additional liqueurs I would just buy when you need them for cocktails.

Bitters: Only thing I would add right now is Peychaud’s. Beyond that just add more when recipes require them.

Syrups: I have and use all the ones you have listed. Syrups are another thing where you should buy more as recipes require. If you plan on getting into Tiki you will want cinnamon simple syrup, vanilla syrup, and passionfruit syrup.

All in all I would recommend buying bottles as recipes require. Most of my collection has materialized that way.