That's a very nice hat, good find
The official name for it is the Khaki Solar Pith Hat, which was introduced around 1938 as the universal hat for troops in India (the Foreign Service Helmet being retained for all other 'tropical' areas'). As for the maker, Imperial Hatworks of Bangalore were a very well known and major supplier of all types of headgear to both the Indian Army and British troops stationed in India. I believe they were an Indian-owned company (so not really 'Imperial' in the wider sense) and operated from the 1930s through the to late 1950s. I would guess it is indeed an officer's privately purchased example.
The insignia is less easy to pin down. Although flashes on these helmets were common, they are notoriously badly recorded. The Garhwal Rifles is as good a guess as any, and the only way to find out for sure is either luck or tracing the officer in question. However, I would tend to disbelieve the story you were given as it sounds like either a bad case of 'chinese whispers' or just a sales story (i.e. bullshit) to me. It isn't Ghurka or DAK, so I would tend to stick to the old 'buy the item not the story' cliche and dismiss the whole story.
If you want my best guess I would go for an English regiment. I know for sure that the use of a green top fold in the puggaree like that was a well known feature of hats used by the Sherwood Foresters (Notts and Derby Regt), and of course, the Maltese cross shape is a feature of their cap badge. At least one battalion of this regiment used a similar green cross shape as a patch in WW1, so there is a precedent, and green is their regimental colour.
That's my thoughts... apart from the nagging "wish I owned it" one