Pictures Covering Home Theater Speakers
Googling "acoustically transparent fabric" returns a variety of companies that provide materials that advertise the characteristics you require. For example:

https://fabricmate.com/fabric/acoustic-fabrics/
or
https://www.audimute.com/what-is-acoustic-fabric

Transparent cloth is available but putting a picture on it is another challenge. The printing process needs to be one that does not reduce the ability of the cloth to let air pass through it. Painting for example would change the acoustic properties.

The people at audimute build sound absorbing panels with a customizable picture. Maybe they can build a panel with your picture that is transparent.

It would be great to hear how this comes out for you.