Once the mortgage on property 2 is cleared, the intention is for it to be my long-term home and for it to be owned in my sole name.
I'm not sure that you mean that. My understanding is that it would be your long-term home from the outset and that your daughter will no longer be registered as a co-owner when the mortgage clears.
Even if you have a joint mortgage with your daughter, then CGT will not necessarily arise. If she is named on the mortgage but never contributes to any repayments, then if you enjoy the asset by living in it and benefitting from the ultimate disposal proceeds there is a logical argument that you are the sole beneficial owner. And even if she contributed to the mortgage repayments, then such moneys might be gifts or loans rather than the acquisition of an equitable interest in the asset.
Death is a very efficient way of dealing with CGT. On death assets pass to the executors and on to the legatees at market value at date of death - with no CGT liability arising. But in your case CGT would not be in point anyway if the asset had always been your only or main residence. So yes, IHT would be the only issue.