Having this PSD file with a layer named "YOUR DESIGN | 5 x 7", what would be the GIMP command (in Linux) to insert a picture there and produce the final graphic file (JPEG, PNG) that includes it?
Generally there's no hope without a program which can handle Photoshop's smart objects. Many programs, GIMP is one of them, can use the most basic layers of Photoshop, but their abilities get short when one should work with smart objects, adjustment layers etc... Photoshop's specialities which together are the backbone of templates and product mockups.
There's a couple of exceptions. Online program Photopea can interpret incredibly many features which as late as 2 years ago were inaccessible for those who couldn't use Photoshop. So, try at first Photopea!
The other exception is Affinity Photo. It can import Photoshop's adjustment layers and layer styles (no guarantee how exactly), but smart filters are rendered to flat pixel layers.
GIMP can open some simple enough PSDs which also may have smart objects. But no warps nor other effects incorporated to smart objects will work. Looks like your PSD is such simple case. The smart object only skews and clips the artwork to the frame in the base image. The lights and shadows of the base image are applied to the artwork via the blending mode of the artwork layer. The artwork is on the top, so there's also a layer mask. See the next attempt in GIMP:
I pasted a random news photo (a mortar man) as a new layer to your opened PSD. I copied the layer mask from the original artwork layer and skewed the image a little to make it as tilted as the frame is. That step is actually not needed. It's necessary only if the inserted image has long straight lines which must be aligned with the frame edges.
Photoshop and probably also Photopea would apply the pre-adjusted skewing of the smart filter automatically, but GIM doesn't do it. Fortunately the layer mask works perfectly.
The layer mask keeps the mortar man inside the frame and behind the chair and the plant. GIMP doesn't have the right layer blending mode, but mode Multiply seems to work well. I lifted the brightness of the mortar man a little because multiply darkens.
The layer mask was copied by converting it to a selection and the layer mask to my new layer was inserted by using the active selection as the layer mask.