Could a helicopter perform a forward inversion (front flip)?
Yes.  This video at 0:50 shows a heli climbing into a loop, labeled at the top as 1.4 G and 25 knots, so it's still in forward flight when it reaches inverted flight.  The view from inside the cabin at 0:58 confirms pitching without rolling.
(At 1:26 the heli does what's labeled as a front flip at low G's, but that includes a lot of roll too.)
The pilot is Aaron Fitzgerald, in a Red Bull helicopter
with registration N154EH barely visible at 1:44, identifying it as a
1985 MBB BO105 CBS4, classified &quot;experimental.&quot;
Wired magazine describes the crucial &quot;hingeless&quot; rotor head:

The rotor head is milled from a solid block of titanium rather than multiple components, skipping the hinges that typically allow the blades to bounce up and down to absorb aerodynamic forces. Instead, the blades are designed to be more flexible. ... The lack of up and down blade movement at the rotor head limits the “play” in the control stick, giving the pilot more granular control of the helicopter.

The video was published 2019 May 24-25.  It is also available without ads here.
I could not determine the exact date of the flight, but Wired also said it was

a practice session for this Memorial Day weekend’s Bethpage Air Show

so that at least confirms it as shortly before 2019 May 27.