NavList:

I wrote: How about "by time keeper" a method of calculating longitude ..... in respect of the meaning of T.K.

Just to enlarge upon what I wrote above.  The coastline would have been mapped in large part from seawards from the ship and cutters using horizontal octant angles and compass bearings.  The soundings and nature of the bottom would have been obtained using a lead line. 

The problem would have been fitting all that to a latitude & longitude grid.  In 1802, latitude to an accuracy of less than 1nm should have been possible at sea from the Sun.  Without frequent use of camps set up along the shore, longitude could only be obtained using deduced reconning from their most recent reliable known longitude or by use of celestial navigation using their best estimation of time from the ship’s clock and the chronometers carried.  The timekeepers, which would have been carefully rated, would have been running from the last possible place where accurate time was available, they might also have been monitored using lunars.  You’d have to read the log of HMS Investigator fully (which I’ve not done) to discover the extent to which lunars were used.  

Therefore, I’m beginning to feel that T.K. in each grid square on the chart means that the shape of the coast was fitted to the lat & long grid using the ship’s longitude which had been derived using the timekeepers to calculate the ship's position using celestial navigation.  If for some reason only a position based upon deduced reconning had been available, then presumably each grid square would have been marked ‘D.R.’.  Others have studied 18th and 19th century hydrography far longer and deeper than me and might wish to add more learned comments.  DaveP