Looking at the specs from the Datasheet for the DLink DGS‑1100‑05PD it appears to be 802.3af compliant. However, the the D785 is a Class 3 PoE device. If I am reading the specs correctly, the DLink DGS‑1100‑05P only has a total power budget of 8 watts available for 802.3af PoE devices.
The measured power consumption of the D785 results in about 4 watts (6.5 watts with USB) of usage under normal conditions to 4.5 (6.9) full. It needs to advertise its power sourcing needs (Class 3) to the DLink DGS‑1100‑05P during the initial negotiation of the Ethernet/PoE link setup (6.5w - 13w). I would suspect that as if load increases (ringer, Ethernet, call, LEDs, speaker etc) on the phones, power consumption increases, power budget decreases, the phone(s) will decrease in performance or reboot. PoE idle is 3.3w. If the total power budget is 8w and both phones are consuming 6.6w idle, that is 83% of the DLink DGS‑1100‑05P's power budget.
The Datasheet is a bit misleading. The device is powered using 802.3at and has a power budget of 18W. That should be enough to power two D785, but it is not enough to guarantee full power for two Class 3 devices.
On my main switch I see that the switch with both phones connected draws about 14W while the phones are booting and 8W when the phones are idle.
If that is case (18w available) "On my main switch I see that the switch with both phones connected draws about 14W while the phones are booting and 8W when the phones are idle." and
"When I connected 2 of the D785 phones, the lights on both phones turned on briefly, then one turned off and stayed off, while the first phone booted properly" and possibly there is a known issue with the Switch or a firmware/PHY issue. Have you tried upgrading to the latest firmware or looking at Dlink's release notes to see if one exists? There appears to be an A and B hardware rev for the DLink DGS‑1100‑05PD
DLink DGS‑1100‑05PD - Rev. A
DLink DGS‑1100‑05PD - Rev. B
I have Rev. B and Firmware 1.00.016. I saw that a BETA firmware was available, but I did not find release notes for it, and the update cannot be installed remotely (switch needs to enter "Firmware-Update-Mode" without VLAN support and fixed IPs). I will try to install the latest beta firmware tomorrow and see if that changes anything.
Here is a link to the release notes
Powering the phone via the Switch and hard coding it to Class 2 end device would not be recommended considering the resources and loads required and power required for those loads on the phone (ringer, LEDs, etc). This could obviously restrict those features from working as loads increase due to power requirements and restrictions from the Switch.