The DLR arrives at Lewisham at the same time the Southeastern train departs; I realize it's a small station, but will Southeastern wait for the DLR passengers?
Most definitely not, no.
There are frequent DLR trains, take an earlier one to make the connection.
And would there be any advantage to instead taking the Tube to St. Pancras to connect to the high-speed train to Canterbury? It appears the total time and fares are about the same.
Zone U12* LONDON to CANTERBURY STNS fares:
£33.00 Off Peak Return (SVR) Route: Not valid on HS1. Valid any train except those timed to depart before 0930 M-F
£38.30 Off Peak Return (SVR) Route: Plus High Speed. Valid any train except those timed to depart before 0930 M-F
If you travel before 0930, the fares are vastly increased.
Alternatively you can use Oyster PAYG to St Pancras or Lewisham and then a rail ticket from there.
via St Pancras:
Oyster PAYG single fares from Mudchute to Kings Cross St Pancras:
£2.70 Monday to Friday from 0630 to 0930.
£2.00 At all other times including public holidays
(The same fares apply in reverse except the peak fare also applies 1600 to 1900 in that direction)
Return fare from London Terminals to Canterbury Stations:
£32.00 Off Peak Day Return (CDR) Route: Plus High Speed. Valid any train except those timed to depart before 0930 M-F
or via Lewisham:
Oyster PAYG single fares from Mudchute to Lewisham :
£1.50 Monday to Friday from 0630 to 0930 and from 1600 to 1900.
£1.40 At all other times including public holidays
(The same fares apply in reverse)
Return fare from Lewisham to Canterbury Stations:
£23.90 Off Peak Day Return (CDR) Route: Not London. Valid any train except those timed to depart before 0930 M-F