Can a forklift rental be considered part of a Fixed Asset project?

Accounting for PP&E is governed by ASC 360. Assuming you aren't talking about a particular industry that has dedicated rules - you would want to look at sub-topic 10 (General).

Basically, the historical cost of acquiring an asset includes all costs necessarily incurred to bring it to the condition and location necessary for its intended use. So as long as the cost is reasonable (market-priced), and not duplicative, you should capitalize it. An example of a duplicative cost would be to repair something damaged in transit - that doesn't add any value to the asset being created it just brings it back to its original state. Or if part of the warehouse was completely finished, and someone backed a forklift into it, you should technically expense the repair.

Additionally, depending upon the length of your construction project, interest is also capitalizable.

Generally concur with John Cook on this, however, you haven't been super specific as to what the forklift is for. You could get a different answer depending on if it is being used primarily in the construction of the new space vs the demolition of the old space. Most likely you have some sort of Asset Retirement Obligation on the books, and therefore the forklift expense would be a reduction to your ARO and governed by ASC 410.

However, if they are renovating a newly acquired space, then even the demolition costs could be capitalized as well.