First, a quick overview of how planets' stocks of rare items are calculated: In general, Black Markets are the best place to get the rarest 'Mech parts and weapons (this requires that you've paid the pirates for access in the random black market event). Research planets have a special niche in that they carry more rare "Upgrades" (i.e. everything other than 'Mechs and weapons). Star League planets carry somewhat of a mix of the Black Market and Research items (plus a tiny chance of seeing ER / Pulse / Gauss ammo and weapons not available anywhere else), if significantly less likely to have good stuff than in black markets. Lastly there are "Special" planets like faction headquarters and the like that have increased chances of rare weapons and 'Mech parts, with each faction having a different set of rares they're more likely to carry. Davion, Liao, and Marik each appear to have two such planets with faction-specific tables of rares even though only one is flagged as their headquarters. Also worth noting that the Restoration, Magistracy, and Taurian special planets don't use a faction rare 'Mech parts table like the others do, only the rare faction weapon table. And all of these stack, i.e. if two store tags each raise the chance of getting a particular rare 'Mech part, then a planet with both tags will have a higher chance still. Note that this is all based on mining the data out of the game's JSON files, I have no idea if this is accurate or if there are nuances to the store behavior that I'm overlooking.
So, decide what categories/tags you're interested in shopping at, then you want to find perhaps 2 planets in close proximity matching your criteria and bounce back and forth between them to refresh their stock until they sell you what you want.
And now for the visual aid:
This is just a fairly simple Google Sheets document, so if you have basic Excel skills you should be able to tweak it to your liking. Feel free to copy this Google Sheets data and change anything you like, republish it as your own, etc. You have my unconditional permission to reuse my BattleTech research however you like.
https://docs.google.com/spreadsheets/d/1TN0FtTlXCkHNFlFP00JgK1Oyhcqmi9ruViAXytH8ANI/edit?usp=sharing
So, decide what categories/tags you're interested in shopping at, then you want to find perhaps 2 planets in close proximity matching your criteria and bounce back and forth between them to refresh their stock until they sell you what you want.
And now for the visual aid:
This is just a fairly simple Google Sheets document, so if you have basic Excel skills you should be able to tweak it to your liking. Feel free to copy this Google Sheets data and change anything you like, republish it as your own, etc. You have my unconditional permission to reuse my BattleTech research however you like.
https://docs.google.com/spreadsheets/d/1TN0FtTlXCkHNFlFP00JgK1Oyhcqmi9ruViAXytH8ANI/edit?usp=sharing
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I suppose that makes Liao useful for something after all. 
