B767 Captain reported receiving an aft cargo overheat message and erratic indications on the attitude director during cruise. Flight diverted and landed safely.

Narrative

Our Inbound aircraft had SATCOM written up as INOP. A chat with Maintenance Control in preflight about status and plan to fix was inconclusive as Maintenance Control was unable to reach HMC (Heavy Maintenance Check) via phone. Upon arrival at the a/c SATCOM issue was cleared per Maintenance Release and we departed as scheduled.Around 1:30 into the flight we received an Aft Cargo Ovht EICAS message. We accomplished the QRH procedure and called Dispatch and Maintenance Control to ensure no issues existed with proceeding ETOPS. 1st call via SATCOM failed. A second attempt was made that was successful. All agreed that Cargo Heat issue doesn't present any problems going ETOPS. Due to concerns about the previous write up and issues with 1st SATCOM call; we asked Dispatch to call us via SATCOM and we would call them as well to ensure SATCOM was operational prior to airspace entry (random route of tracks). Both calls were successful and flight continued as planned. Later in flight; while I was on my rest break; I received a call from FOs saying the lower amber band on the CA EADI (Electronic Attitude Direction Indicator) was fluctuating wildly and that when steady was as much as 30 kts higher than FO side. We discussed and agreed they would call via SATCOM to discuss with Dispatch and Maintenance Control. A few minutes later I received another call saying SATCOM was inop and they couldn't reach dispatch. I returned to the flight deck and began the process of calling ZZZZ [center] to get a patch that way. We tried multiple frequencies to establish contact with no response on any. We queried ZZZZ1 [center] for any suggested frequencies; they returned the same list as available in our manuals and an another attempt was made to contact ZZZZ; again to no avail (no issues contacting ZZZZ1). During the event; we were No Comm on the ACARS and the a/c WiFi was not working for either of the flying pilots. Occasionally; the Relief Pilot was able to briefly get data. We utilized other company a/c to pass messages to dispatch on our no comm and aircraft status. As a crew we discussed the threats (see below) of the situation we faced and determined it prudent to divert to the closest station with acceptable weather conditions. ZZZZ [airport]; ZZZZ1 [airport]; ZZZZ2 [airport]; ZZZZ3 [airport] were discussed as options amongst the crew. At times while we had limited WiFi coverage; we attempted to contact dispatch via personal devices. From these limited opportunities we received a message from dispatch that ZZZ3 [airport] was not preferred due to slot issues and that ZZZZ4 [airport] and ZZZZ5 [airport] were both viable choices. After a further discussion between crew members; a decision to divert to ZZZZ4 was made. Coordination was accomplished with ZZZZ1 and we proceeded direct ZZZZ4. Upon selection of ZZZZ4 as an alternate; ACARS loaded an incorrect destination in the initialization page and even when VHF coverage was regained; we were unable to send ACARS messages outside the a/c. We did receive messages sent by dispatch and could get wx info. An uneventful approach and landing were conducted. After landing; phone calls were made to Maintenance Control; Dispatch; Chief Pilot; and Supervisor to facilitate next steps. An assessment by all pilots revealed only the Relief Pilot had an uninterrupted rest period. The flying FO remained on the flight deck for the entirety of the flight and I returned to the flight deck shortly after my rest break started. A further discussion amongst the crew determined that continued flight beyond ZZZZ4 was not in the best interest of safety as fatigue from the lack of rest and event became evident. A rest break was coordinated with the Supervisor and the flight crew departed the airport. During the event; all crewmembers looked through the FM (Flight Manual) to determine; in the absence of an opportunity to receive input from Maintenance Control/Dispatch; whether any procedures or information was available to helpmake a more informed decision. I reviewed the unreliable airspeed QRC procedure and found it focused on IAS disagreements which did apply to the situation we saw. A review of various FM systems information yielded no clarity on how that data was calculated nor its impact on operations. At our cruising speed; we could not see the lower sections of the airspeed tape and were unclear as to where the current stall speed was depicted on the CAs EADI speed tape. Discussions regarding the possibility of erroneous stick shaker activation; stall warnings; and stick nudger activation at lower speeds that might be within the erroneous CA Amber band. This uncertainty of reliable airspeed data/calculations on the CA flight instruments indicated to us that the safest course of action was to divert to the nearest station with acceptable weather. That would allow us additional fuel to better contend with any unforeseen consequences related to the unreliable/erroneous EADI data. In the brief opportunities we had to assess wx at a variety of coast in airports; we noted that many were low IFR. We assessed the threat of unknown airspeed issues predicated an airport with VMC conditions; thus ZZZZ3; ZZZZ4 and ZZZZ5 were identified as good candidates. After being advised ZZZZ3 was a poor choice by Dispatch; ZZZZ4 became the preferred option. The entire event was complicated by the failure of SATCOM. The inability to contact Maintenance Control/Dispatch was both a challenge and a distraction. After multiple failed attempts to reach Dispatch; via SATCOM; we requested a clearance to ZZZZ4 from ZZZZ1 and [requested priority handling]. Once the divert decision was made; the crew focused on flying the aircraft and ensuring a safe approach and landing. The decision was made to slow and configure early to identify any unexpected issues as soon as possible. The threats mentioned above were covered. On arrival into ZZZZ4 ATC advised that we could not comply we the speed restrictions on the arrival and were granted relief. During the descent; amber band indications began to return to normal and by approximately 10;000 feet; both EADIs reflected similar data. The a/c was fully configured early and no unexpected issues arose. A non-eventful approach and landing was completed. CFR (Crash Fire Rescue) equipment was not requested; but was present and followed the a/c to the gate. After the flight I also spoke to the Maintenance Control and avionics reliability engineers regarding issue to help diagnose the issue and affect a repair.Important lesson learned… ZZZZ Radio provides suggested frequencies based on geographic location. The best frequencies for our location contradict the rule of thumb guidance for frequency selection in the FOM In our location; the frequency suggested by the Airway Manual was ZZZZ; given the time of night; we attempted contact on three frequencies. Those were also the frequencies provided by ZZZZ1 Radio when queried; yet we received no response on any. An email to ZZZZ Radio after even mentioned that the lower frequency only works in a certain country. A chart that reflected optimal frequencies with geographic distinctions would have provided clearer and more easily found information.The support and input from both FOs was appreciated and valuable. Their contributions were highly effective; and helped me in my decision making process. Dispatch continued to provide support as best as they could without the benefit direct contact and without any feedback from our side. Unfortunately as the SATCOM system wasn't working correctly; we wouldn't always recognize when message had been sent. We did not receive EICAS alerts for received messages (another previous flight write-up and possibly linked to the SATCOM issue).