For crafting non-magical items.
From the Downtime Activities on page 187 of the PHB:
Crafting
You must be proficient with tools related to the object you are trying to create (typically artisan’s tools). [...] For example, someone proficient with smith’s tools needs a forge in order to craft a sword or suit of armor.
So RAW it would appear that crafting doesn't actually require any ability checks. You just need proficiency in artisan's tools to be able to craft non-magical items in the first place.
But the DM might call for other ability checks that require artisan's tools.
From page 154 of the PHB:
A tool helps you to do something you couldn't otherwise do, such as craft or repair an item [...] For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver’s tools, or a Strength check to make something out of particularly hard wood.
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.
Best Answer
The only way I can find relies on Unearthed Arcana material, and requires that you not have proficiency with smith's tools, so it may or may not be useful to you.
The Artificer, from Unearthed Arcana, gets this feature:
Since Unearthed Arcana isn't designed to be multiclassed, we don't know whether you would gain tool proficiencies when multiclassing into it. Luckily, we can choose the Gunsmith archetype, which gives us this feature:
So assuming your DM allows you to use Unearthed Arcana material, and allows multiclassing with it despite it not being designed for multiclassing, 2 levels in Artificer will get you expertise in smith's tools.