you may need to delete the message entirely, because everytime the message with the photo that is on the messaging/phone provider is probably resend back.
when you delete from gallery you are only deleted from a downloaded/cache on phone not the text message or server that resends.
It depends on the source of the app that you saved from might need to save to folder or use share to email.
try esfile manager app, i was able to even copy files to desktop by using the wifi network transfer (note must be on same wifi account example like home wifi)
what happens when you switch to another sms message reader such as gosms?
are there no options when you double click or long press, each sms app has it's own way to download