The project is based on his idea to start with.

你这个句子to start with属于多余，语焉不详, 应该删去。谓语 be based on已经说明了该项目是基于他的想法。To start with可以作评注性状语，通常位于句首，逗号隔开。偶尔也可以位于句末。但这不属于悬空不定式。

To start with，the project is based on his idea.

The project is based on his idea, to start with.

The project is based on his idea.

该书告诉你了，悬空不定式的使用不是随意的，而是用在某些特定结构中，即该书所说的六种结构：a, b, c, d, e, f, g六种情况。除了这六种情况，则不宜滥用悬空不定式。