How do we re-measure long term Intercompany accounts and long term loans?

I'm sorry, "re-measure"? I don't understand what you wish to do.

if it's FX translations, since it's probably cash, then the exchange rate on the date.

Look at FAS 52 CAS 380 I believe

Since our functional currency is USD, we need to re-measure the local currency book based on the temp method . Monetary and non monetary accounts have different treatments. How do we re-measure long term Intercompany accounts and long term loans?

In my understanding, the short answer is, for a dollar functional subsidiary, there is no OCI currency translation account. All translation g/l flows through the income statement, not balance sheet. Most companies have subs that are local currency functional, so a dollar functional sub tends to be a rare situation that people are not as familiar with.

Translation accounting has always been a confusing subject with part of the blame attributable to poorly explained terminology in FAS 52, in my opinion. For example, people confuse current method with current rate method because the names are so similar.

More specifically for intercompany accounts:

a) If an intercompany loan is considered short term, exchange gains or losses are recognized in the income statement, under all functional currency situations.

b) If the i/c loan is long-term, the exchange gain/loss would be included in the equity section under the current method or the income statement if temporal method is used.

It's the part b rule that confuses everyone. Since your sub is dollar functional, you must use the temporal method which means all re-measurement (translation) gains or losses go directly to the income statement.

So the end result is for a dollar functional sub, g/l on all i/c accounts go to income statement along with all other g/l on all other accounts.