Hello Dougie,
I wonder if Ballarat Place, Dollar is a House rather than a Street.
On the 1891 Census, Dollar, there are several families living in Achilton Road, Dollar (Achilton may be an Anc**try transcription error for Ochilton) However, 2 families live at Balariet Place, Achilton Road. Agnes Thomson, age 46, (Household Schedule No. 68) & Jessie Harvie, 49, (Household Schedule No. 69) 1881 Census, Jessie Harvie's address is Ballarat, Dollar. 1901 Census, Jessie Harvey's address is Balarat R, Brickwork Rd., Dollar.
Ochilton Road (now named Devon Road) runs south from Bridge Street, past Ochilton on this OS Six Inch, 1888 - 1913 Map.
http://maps.nls.uk/geo/explore/#zoom=17&lat=56.1619&lon=-3.6762&layers=6 Slide the Blue dot on the bottom left of screen for an up-to-date satellite over-view. Or, click on Side by Side towards top of screen. You'll notice most of the streets in Dollar have been renamed!!
You can also change the maps by clicking on 'Select a map/map series' on left of screen. The OS Six Inch 1843 -1882 Map is very similar to the map which 'MonicaL' posted. (Just love the nls site)
The other reason I thought Ballarat may be a house (2 at least) is from information about the Dollar WW1 War Memorial.
www.warmemscot.s4.bizhat.com/warmemscot-ftopic5780.html The 2nd name on Page 2 is Robert Sinclair Dickson, 6th Battalionn, Australian Infantry. Son of James & Georgina Sinclair Dickson, of "Ballarat," Dollar, Clackmannanshire.
3rd name, Page 2 is Peter Dudgeon, 4th Battalion, Seaforth Highlanders. Son of George & Elsie Dudgeon, of Ochilton Rd., Dollar, Clackmannanshire.
4th name, Page 2 is Robert Eadie, 9th Battalion, Cameronians (Scottish Rifles) Son of Mr. & Mrs. Robert Eadie, of Ballarat, Dollar, Clackmannanshire.
Both Ballarat & Ochilton Road mentioned but not together, however, there was a Ballarat in Dollar as late 1918/19. Almost within living memory. Someone in Dollar will have the answer to this mystery.
Regards, Dod.