I suspect most companies use F1 for he -700 and F5 for the -800. As you say 18K plus a good 50+ on the TASS but the kick depends on what you set the climb thrust to. CLM-2 probably would be ok. My last outfit had an SOP of not derating climb thrust but on hte -700 we mostly still did it as the kick is pretty uncomfortable for the pax.
The thing with these things is this. What is most likely cause?
With the known gotcha of the Gross weight/ZFW in thje FMC and this flight probbly needing in the regon of 10T to go home, the result is exactly as seen. Well below actual V1 so if they'd abandoned the takeoff and slowed to vacate they'd have been only halfway down the strip... Hence my earlier comment.
Either way, it's not the first or last time these things will happen and as said, nice pitch control and a safe outcome in the end.