Charging brick

Yes, batteries are allowed in carryon. In fact, if it's a lithium battery, it MUST be in carryon and not in checked luggage.

Is what you have a Li-Ion battery that you use to charge other devices?

If so, read the rules for Li-ion batteries.

I am not familiar with Mophie products but see there is a variety of items they sell.

It will recognize "litlhium battery."

TSA doesn't care about your battery, it's going to be the airline that determines what you can bring. But yes, IN your carry on (and not checked) is a requirement, and you have to confirm the max watt hours w the airline.

TSA doesn't care about your battery....
See TSA "rule": https://www.tsa.gov/travel/security-screening/whatcanibring/items/lithium-batteries-more-100-watt-hours

Lithium batteries with more than 100 watt hours

Carry On Bags: Yes (Special Instructions)

Checked Bags: No

Lithium batteries with more than 100 watt hours may be allowed in
carry-on bags with airline approval, but are limited to two spare
batteries per passenger. Loose lithium batteries are prohibited in
checked bags.

FAA "rurle" for carry on baggage batteries: https://www.faa.gov/about/office_org/headquarters_offices/ash/ash_programs/hazmat/passenger_info/media/Airline_passengers_and_batteries.pdf

Lithium ion batteries (a.k.a.: rechargeable lithium, lithium polymer,
LIPO, secondary lithium). Passengers may carry all consumer-sized
lithium ion batteries (up to 100 watt hours per battery). This size
covers AA, AAA, cell phone, PDA, camera, camcorder, handheld game,
tablet, portable drill, and standard laptop computer batteries. The
watt hours (Wh) rating is marked on newer lithium ion batteries and is
explained in #3 below. External chargers are also considered to be a
battery.

yes, and the airlines have FINAL authority, even if all IATA/DGR rules are followed. they can, and do, say no. not as prevalent in the US, but it does happen.

Here's why its better to have the Lithium batteries in the plane cabin and not the cargo hold:
https://abcnews.go.com/GMA/News/video/phone-battery-bursts-flames-flight-56983511

yes, exactly.

Nobody asked at security, but our Mophie has gone on several trips.