Would it be Bid’ah to do Salawat,say la ilaha illallah, say Subhanallahi wa bihamdih, say subhanallahi al-’adhim and to say Astaghfirullah a few hundred times every day and in that fixed order?
Any religious practice that cannot be traced back to Muhammad (p.b.u.h), supported by the Quran, Hadith (where weak ahadith may be included) or assumed to be continuous tradition from the time of the prophet, is bid'ah.
Only Salafi reject bid'ah categorically.
According to Hanafi, Maliki and Shafi'i madhab, it is forbidden to alter any practice that goes back to the Quran or reliable traditions (in particular, jurisdiction and anything concerning congregational prayer and Salat), whereas an additional practice that is based on Islamic faith and not contradicting known rules is considered halal as long as it is a voluntary choice of the believer.
Hanafi and - even more - 12er Shia accept also consent of scholars per analogy or extrapolation to be declared compulsory.
Perpetual prayer practice as you asked, is allowed according to the majority of Hanafi, Maliki and Shafi'i scholars. It's common practice in Sufism and Alevite Shia; I don't know how it is handled by 12er or 5er Shia. Most modern Hanbali scholars follow Salafiya to larger or lesser extent and reject such practice.