Should we make an effort to limit organic growth for a Beta?

Don't focus on the quantity but in the value you provide. Numbers ca confuse you to think you have "high demand" in beta, to later launch and fall short. Or launch with no beta traction to later find your core users and go viral.
The metric is not in the numbers but in the value you provide. As Paul Buchheit said "is better to build something a small number of users love than something a lot of users like"

For Alpha / testing round - yes. For a Beta - absolutely not.

If you get lucky and your app 'goes viral' then this is a good problem to have. Odds are that you will able to sort out any issues relatively quickly, whereas artificially limiting growth would potentially result in losing out on a lot.

As Reid Hoffman famously put it - "If you are not embarrassed by the first version of your product, you've launched too late."

I would say that there is no need to throttle your growth while in beta. That being said you should definitely have ramp up strategies, and predefined metric levels to know when to execute which strategy. It is critical to your brand to make sure that you can keep up with demand. Too much demand is a good problem to have until your servers go down and stay down.

I assume you mean "Inorganic Marketing" versus "Organic Marketing" than inorganic growth. While, organic marketing is always a right way to move forward for any startup, it shouldn't stop you from adopting inorganic means as well. If testing is all that justifies the intent then you should stick to the organic means to do the same. Test, Measure, and Iterate to build a better version and then chose inorganic methodology to attract people.

Inorganic may be tempting at times but, you should try to visualize beyond gory numbers. More "Like" doesn't means more customers.

What do the Large Corporations do when they release their games? Battlefield - Call of Duty

These companies work in the same field as Android Mobile -

The Operating System is the same, but the hardware is different amongst users. Unless your game is for an IPhone - You need as many beta users as you can get to make sure you can work out all bugs across multiple types of hardware.

Best way to do this is set a Set time registration - 5 days open for the Beta - the more you get the better but here is the secret - you need to get the Beta testers involved - create a community - This has so many benefits

for example you can save months in costs related to testing - as you literally have people working for you for free.
Working together they will provide you a huge insight into what you can improve on and if you actively engage with them they will stay by your side as you complete the app. They will feel like they played an important part in the application and will want to share their work with everyone they know.

Hope this helps