Everything is going to depend upon the exact wording of the tenancy.
"The storage is now a commercial building, still owned by them but the hirers..."
From what you say, it appears that the charity are now a landlord and are renting the storage building to tenants?
"It specifically states it is for the sole purpose of accessing a storage room."
In this case the charity will still have a need to access the storage building from time to time as a reasonable landlord in order to do any necessary checks etc.
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"...the hirers are not allowed to use the side doors except to follow the fire escape route"
I'm a little bit confused by this. Who or what is saying that? Do the side doors open directly onto your rented property?
Also, depending on how the easement in your tenancy is written then the tenants of the storage building may well have the right to use the path from the side doors (will also depend on what their tenancy agreement says as well).
Things can all get very tricky indeed.
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But I would just like to return to the title of your post for a moment. 'How to solve this without conflict?'
You mentioned:
"I'm now being told by one of the individuals connected with the charity that I cannot store anything in front of the gate"
Looking at your diagram (thank you for by the way - it really does help) I would agree. However that does not mean that you cannot use the rest of the path at all.
If the charity can still use the right of way as conveniently and practically as before then you can use the pathway yourself.
So, for example, if you had an outdoor barbecue and perhaps some chairs and a table on the patio and the barbecue was placed so as to stop the gate from opening or block the entrance then you could not do that.
In contrast, if the barbecue was not blocking the gate but was further along the path and so the charity needed to walk around your chairs and table on the patio and your barbecue further up the path then that would likely be okay.