Bluetooth bee and different profiles
I haven't worked with this Bluetooth Bee, but I'll attempt to provide some insight.

In regards to HID:

It's unlikely you'll be able to support HID onto the BT Bee; in your datasheet, the Bee supports version 2.0 with Enhanced Data Rate (EDR), whereas according to the HID Spec, HID conforms to 2.1 + EDR or better. Also, HID is supported by L2CAP, so you would have to go low level to support it yourself (which is a great feat in itself). If you are dedicated to HID, find a module that supports it natively.

This RN-42 from Sparkfun for instance:
https://www.sparkfun.com/products/10823

If you're interested in the HID Spec...

In regards to connecting issues:

I suspect that the issue may be occurring because you may have your module configured to be a master device and you're trying to initiate the connection from your computer. The device initiating connection is always the master device. Your datasheet for the module mentions that you can configure your module to be a Slave.

In regards to SDP:

The datasheet doesn't say anything about the Service Discovery Protocol (SDP), which is a convenient Bluetooth feature in which a random 6 digit number is generated and compared by two devices. In this case, your module supports something known as Legacy pairing, where a 4 digit PIN in chosen and/or entered by the user. 

Your computer very likely supports SDP, however when it attempts to pair with a device that does not support SDP, the computer should utilize Legacy pairing to conform to your module. In short, you don't have to worry about SDP.