Elemental Wild Shape just grants access to different forms for you to Wild Shape into. As such, it's limited by all the restrictions and gains all the benefits of normal Wild Shape.
To answer the second half of this question, we need to know if Beast is a keyword, or if it's simply a general descriptor that could apply to anything we Wild Shape into. The answer to this is that yes, Beast is a keyword monster descriptor.
However, that is thankfully irrelevant. The text under Beast Spells refers primarily to Wild Shape, and has ancillary text that refers to Beast Shape. However, based on the text in Wild Shape, I believe that they are used in the text interchangeably and thus should be applied interchangeably for the rare case where you are not in a Beast form.
So yes, you can cast spells while an elemental using Beast Spell.
Rule interpretation in general
There are two points to remember when looking at an ambiguous rule:
Your group, with the DM as final arbiter, decides the ruling for the rule. So, you need to establish what this is for each group that you play with.
Specific beats general. Start with the most general rule you can find and see how the specific rules change this.
Armor resizing in particular
For your issue, start with PHB p. 145 (ignoring the optional part because it makes no difference to the case at hand)
Variant: Equipment Sizes
In most campaigns, you can use or wear any equipment that you find on your adventures, within the bounds of common sense. For example, a burly half-orc won't fit in a halfling’s leather armor, and a gnome would be swallowed up in a
cloud giant’s elegant robe.
There's your general rule: no armor that fits a PC race (size S or M) would fit a bear (size L).
Wild Shape (p. 67 PHB) says:
Worn equipment functions as normal, but the DM decides whether it is practical for the new form to wear a piece of equipment, based on the creature’s shape and size. Your equipment doesn’t change size or shape to match the new form.
No help there; the armor will be too small and the wrong shape. In addition, there is a ruling that has to be made right here: is it practical for a bear to use armor even if it is the right size and shape? My ruling would be yes but your results may differ.
Now from DMG, p. 140:
In most cases, a magic item that's meant to be worn can fit a creature regardless of its size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer.
The ruling needed here is does the hide armor a) need easy adjustment or b) magically adjust itself. If b) then all is easy, if a) then the druid may need to allow it to drop to the floor and then get some assistance to put it on which would take 5 minutes (PHB, p. 146) which makes it pretty useless in combat.
Those are the issues; your group needs to make the ruling.
Best Answer
It would work while Wild Shaped, if you transform into a creature your DM sees as suitable.
As it is only an insignia, it could be attached to a cloak that should be able to be worn in most forms. It would depend on your DM, however, as the PHB states under Wild Shape:
If the DM allows it on a cloak that remains worn, the bonus would affect your attacks as they use natural weapons. But if they rule it wouldn't be feasible to be worn (such as if you transformed into a wasp), it will be merged in and no bonus will apply.