What size chessboard?

A good rule of thumb for square size vs King base diameter is  King base diameter = 0.75 * square size or 1.66"  = 1.33 * King base diameter 1 1/4"

Your calculation is right. If it does  looks and feel a bit too small you can go for more standard size of 2".

Chess Set Sizing - Pieces vs Board...

The primary sizing criteria for the set and board is that of square size vs King base diameter...

A good rule of thumb is...
King base diameter = 0.75 * square size
or equivalently....
Square size = 1.33 * King base diameter

For example: King base = 1.25 inch => Square size = 1.33 * 1.25 = 1.66 inch.....I would suggest to round up to about 1.7 inch square size (I prefer squares that are slightly larger, versus slightly smaller, i.e., so that the pieces aren't cramped).

Any variation from these ratios, or the size of individual pieces and pawns becomes a matter of personal preference...
http://blog.chesshouse.com/how-to-select-the-right-size-chessboard-for-pieces/

You might also want to consult the FIDE Manual, Sections 2 and 3 on equipment regulations for FIDE events...
https://www.fide.com/FIDE/handbook/Standards_of_Chess_Equipment_and_tournament_venue.pdf

Personally, that photo you posted of your 1.25" base diameter King, on your old 1.625" square veneer board, looks fine to me. I prefer a study board a bit on the tight side to save on desk space. For OTB play, maybe go a bit bigger? It's your call. 

2" square is pretty drastic size up from your old board. Your best bet is to use some blank paper and tape (pizza box works great) to draw a 1.75" square chessboard and a 2" square chessboard. Take your time and make them exact. Spending a few minutes to draw a new size board on paper before having it made is time well spent. 

How is the climate and humidity in Barbados? Do solid wood boards do ok? 

Firstly, though there are many rules of thumb, square size with respect to chessmen size is quite personal. It is also (if you are picky as I am) quite precise--e.g. to me, a board with 55 mm squares has a different feel as compared to a board with 2-1/4" (57 mm) squares, even though the difference is only 2 mm. Furthermore, (for me) it depends on the holistic design of the set rather than key parameters like King height and width--some sets (to me) look better with a bit more or less space even though the King stats are identical.

I second Pawnerai's suggestion to draw chess boards to just try it out before you make a significant investment. I furthermore suggest looking up chessmen of that size on web-shops, and hope that they not only show pictures of the chessmen on a board but also mention the square size; and scouting/begging on this forum for pictures of a set similar in size set up on varying square sizes. (I believe about half of my comments on this forum are a variant on: "Nice chess set! What is the square size of the board pictured with it please?").

If you think your current board size is too small (looks fine to me), I would make a very small adjustment. Even going from 1-5/8" to 1-3/4", as suggested by Pawnerai, is already an increase of 3.2 mm. I likely wouldn't go over 2".

Having built in magnets lets you practice in more positions. I love my 12" magnetic travel board. But I don't have a big table here in this RV.

I ordered a 1.75” board, because I will be ordering pieces that are slightly larger. So I’ll either use the new board with the old pieces, or with the new ones. 

Thanks for the insights!