As a native speaker I would understand either of those sentences, however the first sentence...
'I used my friend's car instead of mine with his approval to be there
is not too bad, but it is a little disjointed so I would say that sentence is not structured in the best way.
The second sentence...
'I used my friend's car instead of mine to be there on time with his
is not as well structured but can be understood.
The third sentence...
'With his approval, I used my friend's car instead of mine to be there
I have a suggestion which may help when determining the correct structure for sentences like these. What will make this a little complicated is that there are several parts to the resulting sentence, which I will explain.
My suggestion is that you need to split the sentence into parts which you will put back together in a structured way.
It doesn't matter which of your sentences you use as they all have the same parts.
- The use of a friend's car instead of own car "I used my friend's car instead of mine"
- It was with his approval
- Reason for using a friend's car (to be there on time)
Now, parts 1 and 2 are relating to the friend, so it would be better to try and keep these parts together to allow the sentence to flow. Part 3 can be either at the beginning or at the end depending what "sounds" correct.
Now you may be pointing out that the first sentence follows this suggestion, but I am saying it isn't structured in the best way. The reason why I am saying it isn't structured correctly is because of the flow of the 1st part into the second part. You start off with referring to your friend and his car, then to you and your car, then back to your friend when mentioning it is with his approval that you used his car. Also, when reading the sentence it could be construed as your friend giving you his approval for you to be on time. You needed your friend's approval to be on time?
A couple of commas would help to structure it a little better by saying
'I used my friend's car instead of mine, with his approval, to
be there on time'
adding emphasis to the fact that it was with his approval that you used his car.
The third sentence is perfect because it flows perfectly. You have your friend, his car, and approval to use it in one part of the sentence.