Have we attempted to experimentally confirm gravitational time dilation?

You don't need a space probe. NIST has measured the predicted general relativity time dilation due to a change in altitude on Earth of one foot!

In addition to specific probes like the one mentioned by called2voyage, the effect is significant enough that it affects everyday operations. For example, the GPS constellation needs regular clock corrections because the satellite hardware sits much higher up the gravity well than the ground hardware.

The Wikipedia page for gravitational time dilation mentions other confirmations in equipment flying in airplanes, and even between two points only one meter distant in height under laboratory conditions.

In general, any instrument taking precise measurements over a distance must take relativistic effects into account as well, especially if it and its target are moving at significantly different speeds or are at significantly different heights. Physics SE may be able to better answer exactly when the effects become impossible to ignore.

Yes, time dilation was experimentally confirmed by Gravity Probe A, launched by NASA on June 18, 1976.

The clock rates of two masers (one on the probe and one on Earth) were compared, and it was found that the difference matched what was predicted with an accuracy of about 70 parts per million.

To address your question of challenges in designing the mission:

To yield an accurate and inexpensive experiment, GP-A required a flight path with a large change in the gravitational potential to provide a large gravitational redshift, and it required a flight path that kept the flight Hydrogen MASER in contact with the ground Hydrogen MASER during data collection. [Stanford]