That means my approach of composing the image from those two is not correct and Apple Photos does not accept it.

I was wrong about the aux image, it is not used in this case. The only reason why my test photo looked so bright was I confused myself with gamma=1.5 and applied gamma=15 which was actually set max acceptable value of 2.5. The same time, having original Apple's JPG with gamma 1.03 and the aux image I cannot get the same look after the procedure of the manual decomposing to "flat" JPEG + the aux image and composing it back, and apply gamma=1.03 then.

Here's a research article what, I suppose, was the source of the idea: researchgate.net/publication/4288952_Display_HDR_Image_using_a_Gain_Map?enrichId=rgreq-0b085e96af6d797348d30128f6a1e31b-***&enrichSource=Y292ZXJQYWdlOzQyODg5NTI7QVM6MzMzMzIyMDQwNDMwNTk2QDE0NTY0ODE1NzYxOTY%3D&el=1_x_2&_esc=publicationCoverPdf

I got a similar result earlier with setting another HDR tag and command exiftool -overwrite_original_in_place -preserve '-CustomRendered=HDR (no original saved)' <image> (BTW it gives that HDR visual tag visible in Apple Photos app, top left corner). It also made photos brighter, but not enough.

That was a good investigation but it is half true. Apple does use an embedded HDR Gain Map, though in combination with the gamma setting. Gamma setting alone makes images brighter, but now so dramatically as iPhone does, with any value of gamma.