I would advise using the new passport on both situations, and to get the VISA on it.
Nowadays you can go or may be even required to go to electronic check-in queues when entering several countries. In other countries you may get faster though immigration using the biometric one.
Official might also do not like the old, non-biometric one. I was using one 3 years ago that was issued in an emergency, and when (re)entering my own country returning from a country known for being easy to get forged documents - and a top of that, the passport registered as being issued in yet another 3rd continent, I was questioned and required to answer in my own mother tongue to trow out initial suspicions I would be carrying a fake passport. Mind you that a decade ago, when I was issued that non-eletronic passport, I was warned not to use it for long, well, for it not being biometric.
Most countries also invalidate the old passport when issuing a new one to any individual, and in these post-computer days your old passport might be flagged as not a valid one, at least in the databases of your own country. When issued a new passport, normally you are not supposed to continue using the old one e.g. you are not supposed to have two active passports from the same country.
There might be a risk having your VISA refused using the old passport. I would use the new one, and would try the normal process and, if necessary, pay any required fees to speed up the process.
Also from a Peruvian travel agency site detailing the national passport legalities: Tramites de Pasaporte Expedicion, Revalidacion y Anulacion
ANULACIÓN DE PASAPORTES: Es automática y gratuita cuando se solicita
un nuevo pasaporte al mismo tiempo.
English Translation: "CANCELLATION OF PASSPORTS: It is automatic and free when a new passport is requested at the same time."