unpacking a ZIP file can put the file on disk and re-create the date/time info
I think that 'mangling' the date/time info is less likely than just copying the file on to the machine (eg from a USB stick) knowing (from the date/time of saving the image) that it could be tied to a time when only the OP was there.
Sounds like someone (manager?) is framing the OP and presumably the only 'evidence' is the date/time info on the computer for that image, and a diary showing who was working that day.
Manager, if bearing some grudge, could easily insert a USB stick, copy the file to the PC hard drive, on Tuesday/Wednesday (whenever she examined it) and make the claim it was found by chance the next day.
Tying file date/time to 'who was in' is presumably going to be 'the evidence' but the fact OP had never seen the image, and downloads are generally direct to desktop, make it look all the more suspicious that it is a frame up job. Unfortunately, depending on how bossy confident the manager is, the directors may just take her word for it and know no better.
It will be essential to put enough doubt into their minds, and if it comes to some internal 'investigation' then ask that you be heard without the manager present. Clearly you need to know what you are accused of (given the vague letter, when it seems only this image was mentioned to you) but you will need to point out that if the manager 'has it in for you' then she could have planted the file on the PC, having found it and downloaded it 6 months prior, on a date only you were working - then if downloaded on another PC and copied via USB stick, it could be placed on 'office' PC at any time.
Given the fact that everyone has the user login details, you may prefer to be vague about whether it was the manager that did it, but put it in more general terms, that anyone who knows a little about PCs could have done this, and as the login details were widely known to staff, anyone else might have copied the file onto the PC. That gets around direct attack on the manager, someone the directors appointed, so their decision comes under attack if you make a direct attack on her (suitability).
(I would not even refer to the login details as 'your' login details, if you can find ways to phrase them simply as 'login details', or if you are pushed, explain how the need for anyone else to access the PC was determined and that in practice it means anyone there can login using 'your' login details.)
I think any 'forensic investigation' is out of the league of a small charity, as they cannot isolate that PC from being used for day-to-day work, especially if it has files for access by other staff via the network.