Usually, the P_t^I is the relative price of investment (or in a planner’s problem the marginal rate of transformation). It belong in the budget or resource constraint, because it tells you how much investment goods cost relative to the numeraire good. Once the investment good has been produced, it adds one-to-one to the capital stock.
However, many models also feature a marginal efficiency of investment shock. That one shows up in the law of motion for capital, because it determines how efficiently investment goods are translated into installed capital. However, that shocks does not have an immediate budgetary impact and therefore does not show up in the budget constraint.