Close range ultrasonic fluid proximity level sensor
First, some physics. 


In dry air at 20 °C (68 °F), the speed of sound is 343.2 metres per second.
At 42 kHz (the operating frequency of the device you linked) the wavelength, \$ \lambda \$, can be calculated:


$$ \lambda = \frac {v}{f} = \frac {360}{42k} = 8.6~mm $$

To send out a "ping" to the target we will have to send at least several cycles. We don't know the response of the transducer but let's assume ten cycles. When the last wavefront is leaving the transducer the first is 86 mm towards the target.

It should be getting clear now that since the transducer can't be in receive mode and transmitting simultaneously that there will be some minimum distance that the device can handle. The manufacturer has stated this to be 130 mm.


  In the first case, are there packages out there that do overcome this, or would it be an option to purchase an individual transducer and hook it up to my own circuitry, optimized for a very low dead zone, ...


You could look for a high-frequency transducer. If you could find one that operated at 10 times that frequency it might work with 1/10 minimum range.


  ... and if so, any examples out there on how to go about building such a sensor out of a transducer?


Search, I'm afraid.


  In the second case, are there any alternatives to a ultrasonic sensor that would help me achieve this goal?




Figure 1. Common methods of fluid level measurement include various float, pressure, and capacitive schemes as well as ultrasonic. Image: Clemson University.

Capacitive would be worth a look.