B737-800 flight crew reported during takeoff; the number two engine exhaust gas temperature indicated higher than normal. Flight crew diverted.

Narrative

Departing out of ZZZ there was an outage of the logbook software program; I proceeded to depart in concurrence with Maintenance Control via voice communications. After printing and downloading the aircraft history; the appropriate code and data was entered to accept the aircraft airworthiness. We closed out at zero fuel weight of 131;890 pounds and 174 souls where before we started taxiing; we configured the aircraft for a bleed off takeoff" and a 5 knot tailwind takeoff and performance systems data. The "before takeoff checklist to the line" was completed before we started taxing with two engine engines. At the end of the runway after the confirmation that the "before takeoff checklist" was complete; I noted that the windsock showed nothing more but a 5 knot tailwind as what we had planned with not much more movement; we accepted the takeoff as throughly briefed. I as the pilot monitoring on this leg verbally confirmed I had the engine failure procedure as the first item on my jepp application in the event that it had to be followed or used. During the initial takeoff run; I noticed that the initial exhaust gas temperature of the starboard side; engine number two; was at a higher than normal value; no exceedance was observed as this is inhibited during the takeoff run to avoid distractions. After rotation and accelerating to a tire speed of 172 knots; the aircraft became airborne with the first officer flying. Rotation was normal but I confirmed with the first officer that we can expect an engine exceedance of sort. At 400 feet the exhaust gas temperatures displayed in red with no warning bells. Temperatures indicated well over 900 degrees of the number two engine. Request of priority handling was verbally stated to air traffic control tower at ZZZ in the efforts to clear the airspace in front of us. There is a charted and known helicopter transition route along ZZZ area ahead of us that I had every intention in the abundance of safety of either clearing or steering away from as we still had both engines producing maximum thrust. After reaching an acceleration altitude of 1;050 feet above fuel elevation; 3;230 feet above mean sea level; thrust reduction began and the turn and reduction of power caused the engine number two to now indicate within limits; barely but it was a step in a direction I could work with. This first officer remained the pilot flying and had the radios. I confirm with him of a speed of 220-230 knots with the current flap configuration of flaps 5 as we were going to come back to ZZZ as this had now turn into a timely threat that did not constituent and immediate return. While I initially requested priority handling and wanted to return back for a landing at ZZZ in the event we had further complication; However after messaging dispatch via ACARS and reviewing the non routine landing considerations; as I anticipated an overweight landing was not going to be a safe option in ZZZ or ZZZ1. Before the ACARS outage occurred; I confirm a divert to ZZZ2 was needed. This gave us time to mitigate the overweight situation and high temperatures and terrain that we were up against. Before climbing out of 9;000 means sea level; I pressurized the aircraft with the engine bleed of the number one engine as that did not experience an exceedance. Air traffic control cleared us to ZZZ2 as appropriate. I then proceeded to call dispatch by Flight deck; Wi-Fi voice; communication where I reviewed our release to the ZZZ2 airport; fuel; en route; alternate of ZZZ3 due to de pressurization and terrain. Apparently; in the time we diverted to ZZZ2 ACARS was down; the Voice communication was required by our dispatch. We diverted well clear of a line of thunderstorms; just east of ZZZ3. We then planned for an overweight landing at 151;898 pounds with an initial landing fuel load of 20;000 pounds of fuel. We accepted and asked for a cruise altitude due to the bleed and pressurization of 29;000 feet to ZZZ2. I contacted dispatch via voice and Flight tech Wi-Fi again where our arrival landing weight approximated 149;498 pounds and a fuel load of 17;600 pounds. We asked for an alternate missed approach procedure out of ZZZ2 for runway XX right in the event of a miss approach and or loss of the engine. For vertical navigation purposes; as we were cleared laterally directly to the ZZZ2 airport; I used a 60 nautical mile ring from the airport at 21;000 feet and 280 knots and a secondary 30 nautical mile ring from ZZZ2 at 11;000 feet and 250 knots.We asked for aircraft fire rescue to follow us to the gate. I assumed the duties of flying pilot landed the aircraft with a vertical speed of no more than 450 ft./min. With a normal touchdown; I briefed the use of the auto brakes one to mitigate the brake temperatures. I also briefed the use of a single reverse and in the event of the need for stopping the use of both thrust reversers. A taxi to the gate with two engines.Exhaust gas temperature exceedance requires further guidance. The use of depressurization; clouds and critical terrain avoidance domestically needs to be addressed. Better connectivity by the flight deck Wi-Fi needs improvement as it took me a while to connect. Even at the gate in ZZZ I had to confidently print the 3 day history due to the system outage. EFB Device charging needs to be addressed from the captains side."