One way to quickly rule out a bad CMOS battery, at least in an otherwise good working computer, is to remove all sources of power, including the battery used for power when used as a portable device. Wait a minute or two, then plug in the power cord only & fire it up. If the battery is bad, the BIOS settings won't hold & you'll know, at best it'll reset to default values. If good, it'll load as normal.
It makes no difference whether or not one's booting from a HDD/SSD or LIve DVD, partition/backup CD, whatever.
there is no data on the HDD which only came with a year's warranty and it passed three months ago
Shame on those HDD OEM's who only provides one year warranty. Until that flood a few years back, 2 & 3 year warranties were a given, premium WD models has carried a 5 year warranty for years. After that flood, I noticed many HDD's that were formerly covered 3 years as only 1. Oddly, I had two of the same HDD, the Samsung HD103SJ (highly popular in 2010/11). The 1st, was bought on the Newegg site & was backed by 3 years. The 2nd, bought on the Amazon site a year later (and less than 48 hours before prices jacked up), had only 1 year coverage, which I discovered only after registring it. I lucked up on the one from Amazon for $49, was still in transit to me when pricing doubled.
It was odd, that out of the 2 identical models, the warranty on the 1st purchased one lasted nearly a year after the 2nd one. Fortunately, both drives are still in use as backups & both are faster. quieter & runs faster than the Caviar Black with similar specs.
Hopefully, yours is still good & this issue can be resolved.