"Let's film a movie at the theater."
Translation:극장에서 영화를 찍읍시다.
You are using the wrong conjugation rule. What you are describing is often known as the Casual conjugation. However the verb 찍다 is being conjugated to the Formal speech level in a Propositive mood. The appropriate conjugation is to add ~(으)ㅂ시다 to the stem:
찍다 → 찍 + (으)ㅂ시다 → 찍읍시다
They are fairly interchangeable. 찍다 is more general-use as it can be used to talk about pictures, video, and documents. 촬영하다 ("to photograph or film") is more technical in that respect. Etymologically, 찍다 is a native Korean word while 촬영하다 is Sino-Korean from 撮影하다.
- 사진 촬영, "photoshoot" or "the act of taking a picture"
- 사진을 찍다, "to take a picture"
- 사진을 촬영하다, "to take a picture", especially during a photoshoot at a studio
- 영화를 찍다, "to film a movie"
- 영화를 촬영하다, "to film a movie"