[RPG] Does the Alchemist’s Lab provide “alchemical equipment” in the sense of Craft (Alchethe)

The alchemist's lab specifically overrides the more general masterwork tool and the masterwork artisan's tools

• Most skills don't need tools. However, a craftsman usually needs different artisan's tools for each Craft skill.
• You can buy a masterwork tool for a skill. The masterwork tool for the skill Craft is masterwork artisan's tools.
• If a specific masterwork tool is available for a skill (and you want that circumstance bonus) you must buy that instead. The specific masterwork tool for the skill Craft (alchemy) is an alchemist's lab.

Below are relevant passages from the Player's Handbook. They don't make a lot of sense when considered individually, which is why I included the above summary.

The Player's Handbook on Favorable and Unfavorable Conditions says

Give the skill user a +2 circumstance bonus to represent conditions that improve performance, such as having the perfect tool for the job, getting help from another character (see Combining Skill Attempts, page 65), or possessing unusually accurate information. (64)

Emphasis mine. The Player's Handbook later describes the masterwork tool as follows:

This well-made item is the perfect tool for the job. It grants a +2 circumstance bonus on a related skill check (if any). Some examples of this sort of item from Table 7–8 include masterwork artisan’s tools, masterwork thieves’ tools, disguise kit, climber’s kit, healer’s kit, and masterwork musical instrument. This entry covers just about anything else. Bonuses provided by multiple masterwork items used toward the same skill check do not stack, so masterwork pitons and a masterwork climber’s kit do not provide a +4 bonus if used together on a Climb check. (130-1)

Emphasis mine. Note that the list of examples isn't exhaustive and that Craft is a skill. The Player's Handbook also describes artisan's tools:

These special tools include the items needed to pursue any craft. Without them, you have to use improvised tools (–2 penalty on Craft checks), if you can do the job at all. (129)

And masterwork artisan's tools:

These tools serve the same purpose as artisan’s tools..., but masterwork artisan’s tools are the perfect tools for the job, so you get a +2 circumstance bonus on Craft checks made with them. (129-30)

Emphasis mine. Finally, the Player's Handbook describes the alchemist's lab, in part, as follows:

This set of equipment includes beakers, bottles, mixing and measuring containers, and a miscellany of chemicals and substances. An alchemist’s lab always has the perfect tool for making alchemical items, so it provides a +2 circumstance bonus on Craft (alchemy) checks. (129)

Emphasis mine. When a skill lacks a specific perfect tool, you buy a generic masterwork tool or, for a Craft skill, generic masterwork artisan's tools for that Craft skill. The skill Craft (alchemy) has a perfect tool available: the alchemist's lab. This makes both a masterwork tool and masterwork artisan's tools unavailable for the skill Craft (alchemy) in the same way you can't buy a masterwork tool for the skill Heal instead of a healer's kit. To get the +2 circumstance bonus for using the perfect tool on Craft (alchemy) skill checks you use an alchemist's lab.

There's no real contradiction here. In general, anyone can use a tool with or without proficiency - if they have proficiency, they get to add their proficiency bonus to checks made with that tool. For example, from artisan's tools:

Proficiency with a set of artisan’s tools lets you add your proficiency bonus to any ability checks you make using the tools in your craft.

However, as you've already discovered, crafting has specific rules:

You must be proficient with tools related to the object you are trying to create (typically artisan’s tools).

So in general, proficiency with tools isn't required to use those tools, and Mearls' answer reflects that, but in the specific case of crafting, proficiency is required.