No, because "la" is a direct object pronoun, and in this sentence you need an indirect object pronoun ("I am going to show the car to her" - "the car" is the direct object, "her" is the indirect object). Indirect object pronouns in Spanish are either "le" or "les", so they only differentiate by number.
Most importantly le is not a direct object pronoun, it is the indirect object pronoun. While direct object pronouns are not used with a stated direct object, they are used with indirect objects with the named indirect object. It is one of the difficult points for learners.
You do not use "lo", because you already have the direct object specified -- "el coche". On the other hand, indirect objects, when present, must be expressed with an indirect object pronoun, and can then be clarified with extra information, thus "le" is required.
- For direct object, you use either the pronoun or the full expression.
- For indirect object, you must use the pronoun and optionally add the full expression.
If anyone is wondering where "se" comes from like I was, it replaces "le" to make it easier to say than two short l words in a row. And the indirect object pronoun will always come before the direct one. Same as if you attach it to the infinitive. "Voy a mostrarselo a el" I am going to show it to him.
"Le" is for indirect objects; it means "him" or "to him", or "it" or "to it" for masculine nouns. In this sentence, it is translated to "him."
In case the term "indirect object" needs clarification: It may be helpful to conceptualise indirect objects as nouns or pronouns that can be worded with a "to" or "for" in front of them: e.g., with this sentence, "I am going to show the car to him."
That is called clitic doubling and is standard practice for indirect object pronouns in Spanish. I was taught that it was required, although I have seen sites that disputed that. Nobody disputes that most native speakers do it as if it were a rule however.
Clitic doubling for direct object is a different story. I don't think it is ever required and it is often just wrong. The only things I have found on the internet about it are.long scholarly papers (which I haven't read) My personal practice is not to use them, but notice the contexts when I hear them.
Your other example has a direct object, not an indirect object. In this example le coche is the direct object and the le represents the a él. As I said, it may not be required, but I think you would be hard pressed to find it lacking in standard written Spanish. It may be disappearing in conversational Spanish, and if so, probably more in some areas than others. So depending on where your tutor is from, s/he may have a different experience than some. But I have to find a language or grammar site that promotes that. This one says MUST.
That really is not any more literal than I am going to show him my car. This sentence demonstrates how Spanish use the indirect object pronoun whether or not an indirect object is named. Obviously it would not be correct to translate both the le and the à Él to English. The à él does show that it is a he and not a she or you, but either construction is true to the meaning.