I've read that Mahasi style noting is faster for getting through the dark night and to stream entry relative to Goenka style, so I'm trying it out. 

I'm finding it somewhat annoying to keep guiding my attention to the rising/falling of my abdomen. While I do make an effort to note thoughts, it sometimes feels like I'm pushing them away in favor of concentrating on my breathing. It feels forced and unnatural. 

I've heard of some techniques where you let your attention flow freely and note whatever comes up. I've tried this, and it feels more natural, though I do find myself getting distracted more easily. I keep my eyes closed and try to stick with noting body sensations and thoughts. 

Any thoughts on whether directing attention back to my abdomen vs. letting it flow freely is a faster path out of the dark night to stream entry? 

There is no "path out of it all" but rather "path into THIS ALL". Give up the idea that you need to get through something from here to there. Better yet, keep noting it again and again as it arises; wishful thinking, desire, aversion, unpleasant, anticipation, ...

I would suggest Freestyle Noting Aloud with Eyes Open and gently fixed on a small kasina object to aid in concentration while noting. Minimum 45 minutes daily session. Keep noting at least 1-2 matter fo fact sensation per second and do this for the entire sit. As Shinzen say, "use the voice". If you have much resistance to it then shout out the label so you realy can hear yourself but do not break the stream of noting throughout the entire session. 

During thesession, every once in a while scan the whole body gently and see if parts are in tension. If so then relax these parts (usually, hands, arms, fingers neck, legs). 

Here is how I go about it (based on Kenneth Folk and Shinzen teaching on Noting Aloud)
https://www.youtube.com/watch?v=fcoN5atRTOI

Excellent advice by Papa Che Dusko - let it flow freely.

At some point during DN and especially EQ you might want to drop outloud verbal (or mental verbal) notes alltogether and switch to pure noticing. But it is a matter of what works for you - some people stick to noting throughout, and some people (like me) maybe better progress with pure noticing.

A friend familiar with the Mahasi system and its genesis tells me the following (I dont even know how to fact check this).

During Burma's colonial past, the church started to aggresively convert people to christianity. It was a very strong conversion drive with targets and bags of goodies targeting the poor. The Sayadaw's decided to fight back. They fought back with the only tool they had ... awakening ... a tool that simply cannot be argued with. Every monsoon for a period of three months they would shove hundreds of people in associated monasteries around Burma teaching (what is called) the Mahasi technique. The sole purpose of the Mahasi technique is to rapidly get yogis to the dukkha nanas. The Sayadaws were confident that a large percentage would simply surrender to the process and pop a cessation. With a taste of nibbana nobody would go looking for Jesus!! The Sayadaws were also super confident that at the end of the three months, people stuck in the dark night would visit the monastery and seek help and that the sayadaws knew how to help. They knew that they could help these people. So basically there would be a couple of 100 people released into the general population who were stream entrants and would spread word of mouth. It was a challenge thrown by the pope in Rome and boldly accepted by the Sayadaws. Basically it was the Sayadaws saying Game On .... heathens!!

The Mahasi technique is weaponized spirituality. But if as a culture you wish to fight a war, or as an individual yogi you wish to get on top of a donkey, brandish a sword and attack the windmill of dukkha, then weaponized spirituality is what is needed.
The Mahasi technique is optimized to get you to the dukkha nanas, it is not optimized to get you out of there.

Please read this series of posts and see if it helps you.

​​​​​​​https://www.reddit.com/r/streamentry/comments/o19kpo/vipassana_the_progress_of_insight_part_1_anitya/?utm_source=share&utm_medium=web2x&context=3

​​​​​​​The beauty of this is that the moment you realize you are distracted ... you are no longer distracted! That's the name of the game - getting distracted, and noticing it, again and again and again, thousands and thousands of times, both on the cushion and off the cushion ... until one day you start to notice that you are both distracted and aware of being distracted AT THE SAME TIME. And then you start to wonder - what is a distraction and what is not a distraction? I thought I wanted to keep my attention on the breath, but what was my intention to do that distracting me from? I thought I wanted to let my attention flow freely, but isn't that already happening? How is it possible to be distracted? How is it possible not to be distracted?

All of these questions are predicated on the idea of there being someone in here controlling the process. But that's all it is - an idea, not a reality. The idea is not needed, and without it there is just attention doing its thing by itself, the way it already is but for the thought that it is not ...

Extra credit exercise:

​​​​​​​Just what is "distraction?" Why is it bad? Or is it?