Given a 2004 Civic Hybrid manual with a damaged IMA battery pack (will not hold a charge), is there any downside or danger in continuing to use the vehicle's electricals as normal? This includes things like headlights, cabin climate control, defrosters, radio, etc. Assume the conventional battery is holding a good charge, in this case. This is the generation of motor that does not have a standalone 12v alternator, relying on the IMA/HV system for everything.
According to Wikipedia, the Honda IMA system is peculiar in the way that it has a conventional starter motor as a backup. Thus, you can operate the car with the HV system disabled. I assume this means the car will also work if the HV battery doesn't hold charge.
Note that this peculiarity is not the case with other hybrid systems. For example, Toyota's hybrid system does not have a conventional starter motor. It relies on the ability of the HV battery and the motor-generators to start the engine.
Does the IMA alternator still function? If you are certain that the cause of your problems is the battery not holding charge, and not the alternator not charging the HV battery, I would say you can continue to use the car, albeit with a reduced fuel economy. I would of course see what other options there are. How much does a new battery cost? And how much does a junkyard battery cost? I'm not aware of the durability of Honda's batteries, but at least Toyota's batteries typically last for the lifetime of the car, so even a junkyard battery will probably be fine.
A 2004 Civic Hybrid MT will start and drive around with the IMA system completely disabled (the breaker flipped, emulating a no-HV-battery condition). The 12V charging circuit will not function at all under moderate acceleration or at a stop. At highway speeds, the charging system will keep up, but around town it may not.
Very disappointing that I can't find a simple answer about how long it will run without the hybrid battery, but hopefully my experience will help someone... I have a 2007 Honda civic hybrid approaching 200Kmiles on it. Three years ago the dealer told me the hybrid battery was down to 10%, which was tripping the IMA and check engine lights. The dealer replacement cost at that time was close to $4,000 so I didn't replace it right away. Some places will fail state inspection with a check engine light, but not where I live. I've been driving three more years now, and getting more noticeable but still running. Mileage went down by around 10mpg, which isn't bad (still getting 30-35mpg). Going up a long very steep hill, it's tough to go more than 30mph (put on flashers if needed), but it gets to the top. Still runs at highway speeds, but you can't 'punch it' to jump into traffic. Recently, it's starting to hesitate or stutter (I refer to it as having dry heaves!), as the IMA tries to kick in but can't. Probably the hybrid battery is getting close to 0% at this point. Now I can get a refurbished hybrid battery installed for around $1,800, but the car isn't worth much more than that. I don't know what will happen, but so far it still gets me to work. Otherwise this has been the most reliable car I've ever owned. Gas gauge also stopped working a couple years ago (dealer replacement cost $800), so I'm using the trip odometer instead. It burns oil, but no mechanical repairs at all (other than brakes, tires, etc.). I wish I bought a non-hybrid back then. 10-15MPG gas savings for the hybrid isn't worth it.
I would ask a Honda specialist but without the IMA battery working the car is less efficient and monthly fuel costs are higher. Hybrids and electric cars are unique in that the save on fuel. Have you asked as to replacement costs? Assume $800. Then consider the miles a year and costs with a battery and without. For example 12000 miles at 40 mpg = 300 gallons. You know your car and miles you drive. I estimate the battery extends the mpg and could pay for itself in 3 years. Good luck.