Thank you Michael, that is an awesome trick! It will take a bit getting used to entering the constraint first, but it is almost what I needed!
Yesterday I found another solution by combining the line and point scripts into one and just cancelling the creation of the line at the end of the script:

#include "GetPoint.js"

function ConstructionPoint()
{
	var ui = moi.ui;
	var pointpicker = ui.createPointPicker();

	if ( !GetPoint( pointpicker ) )
		return;

	ui.beginUIUpdate();
	ui.hideUI( 'StartPrompt' );
	ui.showUI( 'EndPrompt' );
	ui.endUIUpdate();

	var linefactory = moi.command.createFactory( 'line' );
	linefactory.setInput( 0, pointpicker.pt );

	pointpicker.bindBasePt( linefactory.getInput(0) );
	pointpicker.bindResultPt( linefactory.getInput(1) );
	ui.bindUIToInput( 'bothsides', 'value', linefactory.getInput(2) );
	
	if ( !GetPoint( pointpicker ) )
		return;
		
	//part of Point script here	
	var pointfactory = moi.command.createFactory( 'point' );
		pointfactory.setInput( 0, pointpicker.pt );
	pointfactory.commit();
	
	//Abort line creation
	linefactory.cancel();
}

ConstructionPoint();

This works wonderfully and adds a point at the typed in distance constraint. The only problem is that I'm left with a bunch of "orphaned" point objects. I'm wondering if it's possible to create those points with a specific style or object-name through that script, so that selecting and deleting them is easier.