For antenatal fetal monitoring in a diabetic pregnancy all of the following are useful except:
Fetal surveillance in gestational diabetes:
“Low risk gestational diabetic patiens who achieve adequate control with diet alone and do not develop macrosomia, polyhydramnios or preeclampsia do not requre antepartum fetal surveillance testing before 40 weeks. In fact, the risk of fetal distress in those patients is as low as in non diabetics and fetal well being can be assessed by teaching the patients about fetal movements and asking them to fill up a chart for kick counts. On the other hand, high risk gestational diabetics and patients on glyburide and/or insulin should have antepartum fetal surveillance testing starting at 32-34 weeks of gestation. There is no conscensus as to what is the best test for these patients.
Weekly or twice weekly NST are the most popular. However biophysical profile (BPP) the modified biophysical profile and CST are also used.”
Fernando Arias 3/e, p 449
According to COGDT 10/e, p 315
“Surveillance for fetal well being often begins at 32 weeks gestation in patients with end organ disease using a twice weekly NST or modified BPP done twice weekly by measuring the fetal heart rate an the amniotic fluid volume. A weekly BPP is similarly useful. Women without end organ disease who requrie insulin often begin fetal monitoring at 32-34 weeks. Women with diet controlled gestational diabetes usually begin testing at 36-40 weeks until delivered. Maternal fetal movement monitoring check count using a count to 10 or similar method is recommended for all pregnant
women, including those with diabetes to reduce the stillbirth rate.
COGDT 10/e, p 315
So from above 2 texts it is very clear that:

Fetal kick count
NST – Non stress test
CST – Contraction stress test
BPP – Biophysical score/profile

are done for anteratal fetal surveillance in diabetes.
As far as Doppler is concerned “The current evidence suggests the use of Doppler flow studies in patients with diabetes mellitus who have pregnancies complicated by hypertensive disease, fetal growth restriction or vasculopathy. It is not recommended as a routine method of fetal surveillance”.
Management of High Risk Pregnancy, SS Trivedi and Manju Puri, p 338
Non-stress test
Biophysical profile
Doppler flow study
Fetal kick count