With a \$100-200 budget it's unlikely you can duplicate the functionality of a \$695 commercially-produced product. Your device may be slower, bulkier, noisier, less reliable.
One possible approach is to use dot matrix printer print-heads to drive pins. See example of pins being driven in a vimeo.com video called “Dot Matrix Print Head - Testing”. You might need to replace the pins with push wires so that you could go from a 3-by-9 pin layout to a 4-by-4 or 4-by-5 arrangement. Some fairly old (and bulky) print-heads already contain push wires.
If the print-head pins are driven ballistically they might not be able to work with the high duty cycle that a Braille dot would require. If you use push wires, you could have a cam that moves a latch plate back and forth, alternately latching wires in place or releasing them so they can move up or down. Or you could have a cam that lifts all the Braille pins at once, and a print head that drives some latch pins sideways to catch those Braille pins that that should remain up. Another cam would clear all the latch pins at the beginning of each character cycle.
There are miniature solenoids as shown below with roughly 1cm square cross-section, with a 1.5mm diameter pin that moves about 3mm. (Listed on ebay as “Miniature Electric Solenoid - 9 V - Push Type - 2 oz. Force - 3 mm Stoke”, about \$7.) The duty cycle of this solenoid is 10%, so like a ballistic print head it would require some latch mechanism, as well as a spring or gravity return. You would need to attach push wires to the end of the shaft because the solenoids won't cluster closely enough.