Is it possible to do a PhD in one field along with a bachelor's degree in another field, all at the same time?
The usual disclaimer applies here: the answer depends on your country, your university, your program, etc etc etc. It also depends on you. There are many variables here and not enough of them have been defined, so no one can answer your question definitively.

Let's theoretically bound a few of those variables and do a thought experiment by answering this.

Let's say that for the sake of argument, you are a genius who can not only get done work at an incredible speed, but who can also comprehend and understand all the work in front of you without issue, and this applies just alike to everyone around you who you may have to do any form of teamwork with (thanks O.R.Mapper). Then yes, you can do both of these things at the same time (because the main limitations here are time, comprehension, and energy) - provided you aren't doing them both at a university which has a rule against this kind of thing. As long as you handle scheduling conflicts in a reasonable way, yes. This means: If you have a time where you NEED to be doing obligations for one degree or the other (physics degree = unmissable physics labs even in a theory degree, for example, or exams) and you can make it to all unmissable obligations for both degrees (say they're in the same university or the same area) then yes. However I can imagine quite easily that you will have to resolve scheduling conflicts often. You will probably have more schedule obligations with an undergrad degree than with a postgrad one and it's tough to be excused from those, even if you have a good reason. I'm thinking about things like practicals, scheduled exams, and other things you really can't miss. I'm also mostly thinking about how it was for me in the UK system, and perhaps in other systems scheduling is more flexible, so YMMV.

Unbounding that variable a little: I will state the obvious; a physics degree is difficult. A PhD in biotech is also likely to be extremely difficult. You are a human being. Folks struggle with bachelors degrees, folks struggle with PhDs. I think that any person who is likely to be imperfect (in a totally normal way) is liable to struggle with one of these, not to mention both of them at the same time.

Personally if it was me trying to do both these things at once, I'd be so overloaded and stressed and unable to learn after only a short while that I wouldn't be able to do it. You would have to have impeccable time management, extremely prudent stress management techniques, and you'd never really have a proper 'break' from working or studying because both of these things are full-time+++ jobs. You'd be taking on two of them at once. You'd be subject to the same chances for failure as anyone working two 40-70 hour work weeks. Whether you succeed or fail is a function of how well you handle all of these stresses, and how you allocate the time. And how little you value sleep. I think, but I don't know for sure, that how 'clever' you actually are has less of an effect on your success than the above stuff. You have proven your intellectual capacity already with your qualifications, but a PhD and a BSc at the same time is serious business and even the most clever humans in the world would struggle with it from even a logistical perspective alone. I know that if I tried to even have a part time job during my physics degree, I'd have probably failed. My friends with jobs did fail. I was told by the people coordinating our program that it's this way by design; it's designed to push us to our limits to show us what the research world is like, and to differentiate between good, better, and best. 

And this isn't even stating the most obvious part: When you're writing your thesis, you will have time for nothing else. 

Since my answer has really been a non-answer I will leave you with this:

I didn't do a theoretical physics degree, just a 'normal' physics degree. I had friends in the theoretical program. They still had to do one academic year of unmissable physics labs for 4 hours every week (to demonstrate the principles of basic physics). They still had to sit exams which were unable to be scheduled to a different day/time. They still had to study with us during 60 hour work weeks to understand the material in the months leading up to exams because there were so many of them and because the material was so vast (this applied even to the insanely clever ones). The only real difference between a theoretical degree and a normal one was that you're able to do theoretical physics from anywhere where you can concentrate (so you can do the 'research' part of your PhD mostly from home if your supervisor doesn't ask you to be in the workgroup's office and if you don't need their help to get things done). This is where my 'depends on program' disclaimer comes in; it might be different where you are. In fact, it almost surely is different.

A lot of it will come down to your motivation level and your ability to multitask without fail. Why do you want to do this (it seems like a strange choice)? I don't ask because I'm being snarky; I'm asking because if you really do do this, this is a question you'll be asking yourself in what I can only suspect will be a very short period of time. 

If you want it enough and you're really really really good, sure, go for it. The long and short version: There is no reason why you can't, but there are many reasons why a person would not succeed at this.  

Would I recommend it? No way. In the end, you are a human (even if you are really good) and something will have to give way eventually.