If you're thinking what I'm thinking, try applying this expression to the text property of a text layer:
var start = 240; // number of frames you want to offset by
var t = timeToFrames(x = time + thisComp.displayStartTime, fps = 1.0 / thisComp.frameDuration, isDuration = false);
timeToCurrentFormat(framesToTime(t + start, fps = 1.0 / thisComp.frameDuration), fps = 1.0 / thisComp.frameDuration, isDuration = false, ntscDropFrame = thisComp.ntscDropFrame);
The start variable is the frame equivalent of the timnecode you want to start from, so you don't have to start from "0:00:00:00". If you need to convert timecode to frames, try my timecode calculator (https://robwomack.com/timecode-calculator/), type in your timecode and click the TC button (it toggles between timecode and frames). Or take a look under the timecode in the AE timeline - it shows frame numbers there. If you want to start from zero, make start = 0.