Thanks for your feedback.I'm not saying there are any transverse forces! What I'm saying is that you're using the transverse mass m = M[T] (in normal notation, [itex]m = m_0 \gamma[/itex]) instead of the longitudinal mass m = M[T]^3 (in normal notation, [itex]m = m_0 \gamma^3[/itex]). Your force is along the direction of motion so you should be using the longitudinal mass.
I took the existing spreadsheet and changed the relativistic mass multiplier from [T] to [T]^3. (third-power). With that computation made on some 30,000 intervals for t, the total time sum jumps to a very large value of 7,967.1 days (86,400 seconds/ day). The arc sine function solution is on the order of 539.8 days. Before [T] was changed, the spreadsheet matched the arc sine solution. Both solutions are made at 99.9% of c.