Given that the HS300 is an ELPH, I think it's unlikely that there's a separate clock battery. Quite a few cameras don't have one. But if it does, there will be a separate little compartment in the battery compartment itself. (See: ifixit guide for replacing a clock battery on the A590)

What's more likely to be the issue is that the spring loading on the door has loosened over time, and the battery simply isn't being pressed up against the contacts in the camera firmly enough. I had this issue with my S90. I found the "spring" in the door (there are two metal plates, the one beneath, that's up against the plastic is the "spring"), and did a partial disassembly and bent the spring plate a little bit to make it push the contact plate against the battery more strongly. Trying this, or adding a few layers of paper below the contact plate might work for you to eliminate the issue.

I'm having the same problem with "replace battery" messages on freshly charged battery. It does seem to be the battery compartment door not making sufficient pressure against the battery. So far, I just press the battery against the contacts and close the door and it recognizes the battery. I may have to bend the door spring or use some paper to increase the pressure in time.

I dropped my 310 in a lake, at the dock. I pulled it out, extended the lens, removed the battery and card, left the battery compartment door open, and put it on my truck dashboard in a hot sun. It recovered its ability to take pictures but would not hold the date and time. I used the trick mentioned above. I put a small piece of folded paper between the bottom of the battery and the door and the camera is working. The paper tends to come out when ever I open the door so I may try putting some electrician's tape on the bottom of the battery to serve the same purpose. Hope it still fits into the charger. If not, I will go back to the paper trick. Try it.

Paper and now cardboard (it's thicker) does the trick for me, but I would like to get it repaired. *I just found a USA service advisory update offering free repair or replacement with refurbished camera for the following Powershot models: ELPH 135, ELPH 150 IS, ELPH 340 HS, SX280 HS, S120, ELPH 140IS through the end of 2019.

It says:

Service Notice: Insufficient Contact to the Camera's Battery Terminal.

Update

Thank you for using Canon products.

The purpose of this announcement is to inform our customers of the details of the phenomenon described below, as well as of the actions Canon is taking to address this phenomenon. Because we value the trust our customers have placed in us, we are dedicated to delivering industry-leading service and support for our products. We offer our sincerest apologies to any customer who may be inconvenienced.

Phenomenon

In rare instances, it is possible that units of the Potentially Affected Products listed below may exhibit the following operation errors, depending on usage conditions, due to insufficient contact between the camera’s battery terminal and the battery, even when the battery has sufficient power remaining.

• The message “Charge the battery” appears immediately.
• The power sometimes does not turn on.
• The power turns off during use.

Canon needs to make good on this wide-spread problem--it is a disgrace and will lose many customers. Cameras should be returned for a fix that can be engineered by canon. Or maybe this is a ploy to get you to buy a new camera--If I do it won't be a Canon this time.my ELF SD550 cost a bundle back in its day and now it is useless.