Does a hurricane with a smaller eye have a lower pressure?
I could only find this old reference(30 years old) Typhoon Structure As Revealed By Aircraft Reconnaissance. Part 2: Structural Variability by Weatherford And Gray 1988. 

The conclusion of the authors is that the size of a tropical cyclone's eye does not have much of a observable relationship with the cyclone central pressure(MSLP) and is shown as a weak correlation. Observations reveal that the contracting eye size of a TC on the average is associated with a decreasing MSLP and an expanding eye size is associated with a increasing central pressure. 

So let's look at some actual numbers from that above paper -

These are again averages for the period 1980-82 tropical cyclones that have been classified on the basis of size of the eyewall 

Eye-class                          MSLP(Central pressure in mb)  MSLP Change(in mb/12 hrs)

No eye                             985                            +/-5

Large eye ( &gt; 28 kms)              987                            +/-8 

Medium eye (15-28 kms)             956                            +/-9

Small eye ( &lt; 15 kms)              954                            +/-12


Note from this table how much the MSLP fluctuates over a period of 12 hours  for the smaller eye sizes. Also remember that a vast number of cyclones do not form eyes. 

Word of caution if you are trying to view this table from the operational forecaster's point of view it is very likely an individual cyclone may depart from these average values. And while these data sets have been obtained for the Northwest Pacific Ocean it is likely they hold up for other basins as well.