Absolutely Not
As others have stated, no you can not use a focus (of any kind) while Wild Shaped, even if the animal you morph into has hands, and they are empty and holding the focus. All of the information you posted in the original question provides the answers for you so long as you cross reference them. But a lot of people have missed a crucial part of why it can't be done.
The description of the Druidic Focus says (5e PHB, p. 151):
A druidic focus might be [...] a totem object incorporating feathers,
fur, bones, and teeth from sacred animals. A druid can use such an
object as a spellcasting focus.
This essentially states that you get to create a focus out of whatever you deem worthy to call a focus; pretty neat.
The description of Material (M) components says:
Casting some spells requires particular objects, specified in
parentheses in the component entry. A character can use [...] a
spellcasting focus in place of the components specified for a
spell [but not costly or consumed components]
This is the start of the unravel. Here it says you can use a focus instead of a material component (so long as it has no cost/isn't consumed). So essentially focuses are just fluff/filler for material components that most people over look anyway, but if you don't overlook that stuff it's still a pretty minimal net gain.
Part of the description of the druid's Wild Shape feature says (5e PHB, p. 67):
- You choose whether your equipment falls to the ground in your
space, merges into your new form, or is worn by it. [...]
This portion infers that you get to wear or hold your focus if you so choose when you Wild Shape, which is pretty cool.
The druid's Beast Spells feature says (5e PHB, p. 67):
Beginning at 18th level, you can cast many of your druid spells in any
shape you assume using Wild Shape. You can perform the somatic and
verbal components of a druid spell while in a beast shape, but you
aren't able to provide material components.
This is the crucial spot. It specifically says no material components. If you recall from what exactly a focus is, it is essentially a fancy material component. Meaning it can't be used, not because you can't "hold" it, but because for Game Design purposes they disallowed it.
This is further backed up by the fact that they allow somatic components (hand and body gestures) even if you don't have hands or the appropriate body part to do the somatic component, i.e. a snake making a hand sign. So holding/wearing the material component/focus is irrelevant because it is completely outlawed in RAW.
The druid's Archdruid capstone feature says (emphasis mine):
At 20th level, you can use your Wild Shape an unlimited number of
times.
Additionally, you can ignore the verbal and somatic components of your
druid spells, as well as any material components that lack a cost
and aren't consumed by a spell. You gain this benefit in both your
natural shape and your beast shape from Wild Shape.
Doesn't this seem a bit redundant? It is because it is trying to show you that it is essentially an improvement to Beast Spells. At this point, once you reach 20th level you no longer need the material/focus component. So even at level 20 the answer is still technically no, but it becomes irrelevant because materials/focuses are no longer even needed.
However, this is D&D - so if the DM says, "Well that's stupid, you should be able to use your focus as long as you are wearing/holding it"... Then boom, there you go: the true answer is "yes, you can, so long as your DM handwaves it".
From PHB pg 67
Bullet point number 3 of 'Wild Shape'
"...You can't cast spells, ..."
While in Wild Shape form, you may use any Discipline that does not involve casting a spell using Ki such as 'Fangs of the Fire Snake' , or 'Fist of Unbroken Air', provided you have the proper anatomy to do so; but you may NOT cast any spells, no matter how you gain the ability to do so, as with 'Fist of Four Thunders' or 'Flames of the Phoenix' .
Best Answer
Your interpretation is correct.
The only thing Invoke Duplicity changes is the point of origin of the spell. There is no other text allowing for any other adjustments to how the spell is cast.
The text on Wild Shape is equally clear: "You can’t cast spells." The next clause is joined by "and" but it parses out exactly to the same effect if you split it into two sentences:
"You can’t cast spells. Your ability to speak or take any action that requires hands is limited to the capabilities of your beast form."
Consider that Moon Druids get Elemental Wild Shape at level 10. The art for fire and earth elementals clearly depict hands with fingers, and all four forms have languages which means they're clearly capable of speaking. There's no text in the ability that supersedes "You can't cast spells" from the basic ability, so they still can't even though they have mostly the same bits a humanoid has.
As for balance...
Allowing the character to cast spells while Wild Shaped is the province of a much higher level ability, Beast Spells, that Druids do not get until Level 18. Allowing a character the ability earlier than that (significantly earlier, as a Druid 2/Cleric 2 has Wild Shape and Invoke Duplicity) is definitely unbalancing.
While the character in question may not be a Moon Druid, some folks consider the toughness of some the Moon Druid's Wild Shaped forms combined with Beast Spells (and a few levels later, Archdruid) to be overpowered by itself. Granting a similar capability to a multi-classed character at an even lower level is even more overpowered.