Unfortunately there is really no other options. You can try to completely replace the system with factory parts as the current service bulletin says but time and time again I have still seen the freezing temperature glitch persist so do not recommend wasting your money. There really just isn't another solution other than finding the old TIS limited service campaign information so you can update the ECM and this has been a dead end year after year. For several years we had hoped a solution like our Gen2 kits would be able to possibly get around the ECM glitch but even they cannot get around the problem. Currently the only way around it is the warmup above freezing and restart before hitting 50mph in that first run cycle.
You don't have to let it warm up for 10 min and then restart it before driving. You can start driving it cold and as long as you restart the engine before you hit the 50 mph mark and the engine is now above freezing the glitch won't happen. I know this is not exactly what you are looking for but it may make it a little easier to deal with. As soon as initial engine (cold soak starts only) temperatures are above freezing the problem goes away. Again this is only something you would need to do once it has positively been identified as the freezing temperature glitch and only for that first run cycle.
We do occasionally run across this where the problem is both the failing air system and it is also on a vehicle that has the freezing temperature glitch.
To clear up your confusion about how the code can come on at startup and while driving there are 3 ways the P2445 code can come on.
It is not normally possible to get the P2445 code on startup AND have the freezing temperature glitch as the case requirements are different.
1) The normal reason to get a P2445 code is a failing secondary air pump. The system will only normally be commanded to operate on a cold soak start when the engine has been off for 7hrs or longer and the initial engine temperature is above 40F. This is the normal version the bypass kit can prevent because it stops normal operation of the system. It sounds like you had this or valve stuck open codes before you installed the kit which is why you initially had the codes on startup.
2) A damaged pressure sensor. This is rare for a 4.7L three valve configured system but it does occasionally happen. Normally this would exhibit itself as a P2445 code that comes on at random and not tied to startup, freezing temperatures or speed. It is also the second cuase of a P2445 code that cannot be cleared at all with the first reason beign a blown air pump fuse. Likely not your problem but a possibility if your P2445 doesn't match reasons 1 or 3.
3) The freezing temperature glitch which was documented by Toyota well over a decade ago there was a limited service campaign at the time to fix it. This is an extremely specific case for the glitch and and there is currently no actual fix for. Because of the specific conditions required for the glitch to occur it should not be possible to get the freezing temperature glitch as well as a normal p2445 code (case 1).
I have worked on this problem extensively with active and former Toyota Technicians and can confirm the service bulletin existed as I found it myself in TIS 8-9 years ago but did not know they expire and get deleted. Since then I have not been able to find a copy of the service bulletin. The techs I was working with remember performing the work but at the time TIS was done by CD media and even with help have been unable to track down a copy. The service bulletin is the only known possible way to actually fix the freezing temperature glitch. We try to make sure all of our customers are aware of it as there simply isn't' a permanent solution for it.
We have COMPLETELY removed every component of the system from the vehicle as well as replaced all components of the system with brand new factory Toyota parts and the cold weather glitch will still happen.
I question it year after year and exhaustively come to the same conclusion that confirms the original service bulletin that I saw that states it is in fact just a glitch in the ECM software. I make sure that if they have a customer on the phone with an early 4.7L that they are made aware of the possibility of the problem as well as mention it on all of our product listings. There is also a large write-up explanation here on our codes page under P2445:
https://hewitt-tech.com/codes/ (the text markup isn't as good as it used to be on our new website as it was our old site yet) Please read this if you haven't already.
If you have any specific questions please PM me or call and ask to speak to me directly.
Paul Hewitt