Is there a way to include an owner's Proficiency Bonus to a monster statblock in any way?

Not currently possible, and DDB is sadly being very slow about this.

In fact what they've done for a Battle Smith Artificer's Steel Defender for example is they've added its stats into an action using the snippet box to show the auto-calculated values on the character sheet, using limited uses as a mechanism for ticking off hit-points.

Having recently done for this for a couple of Grim Hollow sub-classes I've copied into homebrew, it's a massive pain to do; if you copy the Battle Smith you can see how they've done it though by looking under the Steel Defender feature iirc.

For summoned creatures from summon fey and similar though the current best option seems to be create a homebrew monster, having calculated the correct values for the character who will be summoning it. But you have to do that for every level they might cast at, so again, massive pain to do.

Otherwise it's just a case of referring to the stat-block as needed, and calculating the hit-point total and keeping a note of it somewhere. I guess that's how you'd do it on pen and paper but it's disappointing when the digital tools could be making it so much easier.

I'm building a companion. They will pass from character to character on an unknown basis. Guess off to Foundry I go.

You could still potentially add its stats into an action like the Steel Defender, but make it part of a feat instead so it can be passed between players? As long as you don't need to calculate anything using a class level this should still work.

If you're interested, here's what the snippet box for the action might look like:

• <strong>Armor Class</strong>: {{12+proficiency}} (natural armor)
• <strong>Speed</strong>: 40 ft.; swim 60 ft.
• <strong>Saving Throws</strong>: DEX {{(2+proficiency)#signed}}, WIS {{(1+proficiency)#signed}}
• <strong>Skills</strong>: [ skill]Perception[/skill] {{(1+proficiency)#signed}}, [ skill]Stealth[/skill] {{(2+proficiency)#signed}}
• <strong>Senses</strong>: [ sense]Darkvision[/sense] 60 ft., passive Perception {{11+proficiency}}
• <strong>Aquatic Assault</strong>: A submerged striker has advantage on attacks against a submerged target without a swim speed.
• <strong>Enraged Strikes</strong>: While you are raging the strike adds your Rage damage to its attacks.
<strong>Actions</strong>
• <em><strong>Natural Weapon.</strong> Melee Weapon Attack:</em> {{(2+proficiency)#signed}}, reach 5 ft., one target. <em>Hit:</em> <strong>1d8+{{proficiency}}</strong> bludgeoning, piercing, or slashing damage.
<strong>Hit Points</strong>:

(you'll need to remove the spaces from the square bracket tags, I had to add these to post here).

You'd set the action to reset on long rest, and then use the Limited Use section to set the hit-points. Where it becomes a pain is if you want it to scale by level, as you'll have to enter each value for every single level. Also please excuse any typos, this is from the aforementioned homebrew since I have it handy, but it hasn't been fully tested yet.

Yeah. It'll be much easier and stronger in Foundry.