They say the power requirement is 12 Watts.
That is quite low for an 8 foot strip and it is possible that it actually draws more.
IF it is 12V rated then that's 1A and a 12V 1A switching power supply would fit in a closed fist.
Whether this is compact enough depends on what size your base is, but many map bases would be large enough.
Measuring a few real world Chinese sourced switched-mode power supplies specifically "designed" to drive LED strips.
70mm x 45mm x 30mm - 1A and 2A units (12 & 24 Watts)
115mm x 50mm x 30mm - 3 / 4 / 5 A units (36 48 60 Watts)
So, the smallest of these would operate two strips of the rating that yours claim to have. I'd expect that that would fit in many lamp bases.
150 LEDS at 12 W = 12/150 = 80 mW/LED.
Actually, that's about right - allowing 20 mA per series stringlet of 3 LEDs = 240 mW for 3 or 80 mW/LED. ie my original "too low" figure is based on what I've seen done, as opposed to what makes sense.