he gave a book to me to read 中to read 是宾补还是？

yes.

不是做定语的，是做补语的，这是个句型。

give sth. to sb. to do sh.

he gave a book to me to read 第一个to 是介词，第二个to是不定式
象这样的句型有很多：

He borrowed the money from me to buy the book.
他象我借钱去买书。

He bought the Hello Kitty gift for the littler girl to cheer her up.

他买Hello Kitty的礼物给小女孩让她开心起来。