No senate missions, no elections. No strange side-effects that I've noticed.
I initially started from this:
https://forums.totalwar.org/vb/showthread.php?t=50446
though I found it better to remove all "former Roman" characters. I've played the campaign game all the way through with this mod and had no problems.
I've since experimented (briefly) with removing the Senate entry completely from the descr_strat.txt file. This requires that you remove the "superfaction romans_senate" line from the remaining roman faction entry and prevents the "faction destroyed" message at the beginning of the game. I've noticed no problems with this though I've not done much testing. Will do so next time I start a campaign.