According to a good old unsourced tree she died on 5th October 1819 in Florence, Italy (edit found also on landed families blogspot (edit 2 contemporary newspaper source at end)
The date seems about right as her will was proved on 24 Jan 1820. https://discovery.nationalarchives.gov.uk/details/r/D139679
Ancestry uk link (subscription )
Found on the way
Husbands memorial at Wimborne St Giles, erected by Barbara in 1811 https://www.geograph.org.uk/photo/3356842
Her marriage at St James Picadilly
(City of Westminster Archives Centre; London, England; Westminster Church of England Parish Registers; Reference: STJ/PR/6/7 St James Picadilly) sharing link
Also recorded along with other family dates in Wimborne St Giles register. Link (Dorset History Centre; Dorchester, England; Dorset Parish Registers; Reference: PE/WSG: RE 3/1 Wimborne St Giles)
Found one Newspaper confirmation of date of death.
British Newspaper Archive
The Scots Magazine - Tuesday 02 November 1819 Register Deaths
Oct 5 At Florence, the Dowager Countess of Shaftesbury