Yes, just like in most countries police have a reasonable amount of discretion about enforcement on many issues. But when/if they get complaints they usually have to act... So avoid having people complain is key, and then having respectable backers if something happens is your insurance...
You haven't said what type of visa you will be on when in Thailand, but I'll assume it'll be standard tourist ( or visa exempt ) . The key is understanding Thai culture and especially that concerning Buddhist religion, and volunteering/caring.
Technically almost anything you do that a Thai could do might breach the 'no work' rule - but if you aren't earning money, or doing something that a Thai might earn money for, then it becomes a greyer area . You would need to have the head monk give you the OK, you would need to be humble and non-ostentatious in your helping, and crucially, you would need to not upset or cause to lose face any of the locals, including Sangha members. Demonstrate your sincerity with a peaceful smile, behave yourself, and you should be fine. Maybe clear it with the abbot that if there is a complaint he will back you up at least to extent that you will just be warned to stop ( because of terrible misunderstanding ) if immigration get involved.
Put your compassion into practice fully and you'll be fine. The law is basically there to protect against harm-doers.
I didn't say you SHOULD do that.... just suggested there's unlikely to be a big issue if you don't happen to have it with you, but have a good copy. Certainly try to keep such a valuable document in protective cover always.
A good photocopy will usually be ok, if the photo is clear and looks like you.... if not, they may come with you to see the original.. depends on the reason they want to see it, and the mood of the officer ( like most of the world )
Once I came in on a two month tourist visa and officer stamped 30 days - probably thought I was on visa exempt. Didn't notice it til next day so had to go to immigration , well-dressed, and explain, without blaming immigration ;) . Fixed. I always check the stamp immediately now. Good luck.