I started writing a comment responding to your last comment and then realized I was writing an answer, so I am just going to write an answer. Putting aside questions about what the most natural way to say this would be, for your given sentence
普段はスプーンで食べるがないから箸で
The only reasonable interpretation here that the subject of the verb ない
is スプーン
.
In the comments, you clarified that what you really want to know is something to the effect of
Are there rules for disambiguating subject/object omission across multiple clauses?
Unfortunately, the answer to this is largely just no. Most speakers work out the content of these omissions based on what makes the most sense, and there are certainly no hard rules that you could apply based on things like particle choice or simple surface level linguistic information. Also, this process is largely the same regardless of the number of clauses in the sentence.
Here is a (somewhat contrived) example:
私は車で行くつもりだったけど、壊れたから電車で行った
I was planning to go by car, but (my car) broke so I went by train
私は車で行くつもりだったけど、鍵をなくしたから電車で行った
I was planning to go by car, but (I) lost (my) keys so I went by train
As you can see, the subject of 壊れた
is presumed to be the car while the subject of 鍵をなくした
is presumed to be the speaker. Unfortunately, judgements like this are made using world knowledge ("common sense") and there typically aren't grammatical markers you can count on. This, incidentally, is one of the main reasons why so many machine translation systems struggle with Japanese.
The fact that in this case the transitivity of the verbs is different is just coincidental; the reason in second sentence could also be something like 酔ったから
.
Edit: some related posts can be found here, here here and here. There are probably more; this is a fairly common topic.