Reference the tool proficiency optional rules from Xanathar's Guide to Everything

You can't apply your proficiency bonus more than once to a single roll, as you've established; it seems like you should probably make this roll using your expertise in performance, since performance is a relevant skill and with expertise that grants you the best modifier. However, there are options to represent your particular combination of proficiencies beyond simply adding more bonuses to the roll.

Xanathar's Guide to Everything, in the Dungeon Master's Tools chapter, includes a section titled Tools and Skills Together, which addresses exactly this situation where a character might have overlapping tool and skill proficiencies. It makes a couple of suggestions about how a DM could adjudicate the situation:

Advantage. If the use of a tool and the use of a skill both apply to a check, and a character is proficient with the tool and the skill, consider allowing the character to make the check with advantage.

[...]

Added Benefit. In addition, consider giving characters who have both a relevant skill and a relevant tool proficiency an added benefit on a successful check. This benefit might be in the form of more detailed information or could simulate the effect of a different sort of successful check.

The chapter then goes on to describe the possible benefits of various tool proficiencies, and gaining advantage on a performance check incorporating that instrument is given as a benefit of proficiency with a musical instrument:

Skills. Every tool potentially provides advantage on a check when used in conjunction with certain skills, provided a character is proficient with the tool and the skill.

[...]

Performance. Your ability to put on a good show is improved when you incorporate an instrument into your act.

An extra benefit to a successful check might perhaps be allowing the bard an automatic success (or at least granting advantage) on a subsequent social skill check to influence the audience after the performance.

The intent of these optional rules from Xanathar's is to make tool proficiencies more valuable, in order to encourage players to take and make use of them as character options, and reward characters who have invested in such specialisation.

Your proficiency bonus can only be added or multiplied once by any ability or combination of abilities

From the Basic Rules / PHB p. 173:

Your proficiency bonus can’t be added to a single die roll or other number more than once. For example, if two different rules say you can add your proficiency bonus to a Wisdom saving throw, you nevertheless add the bonus only once when you make the save.

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue’s Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

Thus, any features that would result in you multiplying or adding your proficiency bonus more than once would not stack - you only get to add or multiply or add your proficiency bonus the one time.

Example: Stonecunning and Expertise don't stack

In the case of your example, here is what they would both do to a roll:

proficient check with Expertise = die roll + ability mod + proficiency*2

Intelligence(History) check with Stonecunning = die roll + Int mod + proficiency*2

Note how both features do exactly the same thing: allow you to add double your proficiency bonus to a qualifying check.

Here is what it would look like if you let both apply to the same roll:

Intelligence(History) check with Stonecunning and Expertise = die roll + Int mod + proficiency*2*2

or

Intelligence(History) check with Stonecunning and Expertise = die roll + Int mod + proficiency*2 + proficiency*2

However you do the math, allowing both to apply to the same roll would result in either multiplying your proficiency bonus more than once or adding it more than once. Either way, it violates the rule and is thus not allowed.

Example: Mastery Ioun Stone and Expertise would stack

A rare few effects add a static bonus to your proficiency bonus. These additive effects are highly uncommon though. Most features multiply your proficiency bonus and thus fall into the previous example and do not qualify. An example of an additive effect is as follows:

The Mastery Ioun Stone says:

Your proficiency bonus increases by 1 while this pale green prism orbits your head.

Since this only adds one to your proficiency bonus, there is nothing preventing it from stacking with Expertise.

proficient check with Expertise and Mastery = die roll + ability mod + (proficiency+1)*2

You still only add your proficiency modifier once to the roll and you only multiply it once and thus they stack validly.