If you were hired by a SaaS company that needed to increase its free user base from 1,200 to 3,000 in 6 weeks w/a budget of $10k, how would you do it?

Here's what I'd do:

1) I'd start by sharing your story on every possible relevant social network in a non-sleazy, non-spammy way. This includes Reddit, Inbound.org, GrowthHackers, and anywhere else you can think of. It's important to understand how those sites work so you approach it in the right way and without coming off as spammy.
2) Next, I'd set up a KingSumo giveaway and offer a liftetime free package to anyone who signs up for the giveaway with extra signups for sharing. Play up the value of getting this service for life, and I'll be the first to sign up. Once you have this set up, reach out to every single internet marketing blog, expert, and Facebook page you can without annoying everyone. Come up with a list of people you think will be interested in the offer and reach out to them. This should get you to 3,000, but if not you can always advertise the giveaway on Facebook, Reddit, Tumblr, and other sites like these (maybe Twitter?). AdWords isn't necessary for this since you'll pay a premium.

I'm pretty sure both of these will get you to 3k while being under $10k, but if not you can always add in...
1) Facebook ads to advertise the free package.
2) Hiring a guest post writer to write posts for you on relevant sites with large audiences. If this is done right, I'm sure you can get to your goal with just this tactic alone so long as you use the right byline to advertise the free package and also target the right sites.
3) Sharing data and lessons is always a good idea. If you have any data/lessons you've gathered/learned from the product you can share those on the blog and then share the post around the web.

That's my quick list. If you focus on a few of these, you shouldn't have any trouble getting to 3k with a $10,000 budget.

I actually work in CRO, so not only am I your prospect I'm also familiar with your problem. I signed up to se how the experience was and although the first step to sign up on the home page was very easy, your value proposition could use a lot of work.

You're competing against a lot of similar services and they all look the same, including yours. There's no clear differentiation of ptengine compared to crazy egg and even Google Analytics, there's also no reason why a visitor should sign up on the page. A user-friendly web analytics and heatmap platform. Implies this is like Google Analytics and I'm left thinking why should I sign up? I don't need more analytics.

Furthermore, does your goal of increasing your free user base from 1,200 to 3,000 only include leads on the home page or people that activate and complete the registration here https://report.ptengine.com/signup/signup_form.htm or people that install your code? Each require a different strategy.

Once you activate, you can remove some friction by auto logging in the user so they don't have to fill in their username and password again from the activation stage or at least carry over the username/email to reduce some friction.

The profile page after logging in leaves you without any guidance on what to do next. Adding the web engage feedback on both those steps might provide some more insight.

Improving and or testing your value proposition and the user experience of signing up will get you some gains but is not likely to get you from 1,200 to 3,000 users in 6 weeks. For that level of growth you'll need to at least do both improve your value proposition and drive the right traffic. For that I suggest partnering with CRO tools like (visual website optimizer, optimizely, convert, etc.) and conducting webinars with their audience. This way you provide the content and they provide the list of targeted prospects who can benefit from your tool. But without first fine tuning your value proposition the CRO audience my be thinking the same thing I am, why do I need another analytics tool? Positioning yourself in the marketplace is key here and it may not be as a "A user-friendly" I can get the same things and likely already are from other "platforms" It's not just a question of value but also a problem of why should I switch.

Hope this helps.