late to the conversation here but in addition to FHR i'd also look for a travel agent who can book using the Impressario (or similar programme) rate - In my experience, these rates are considered a direct booking with the hotel which will earn status and points and similar benefits as FHR (minus the guaranteed late check out). I read from the Luxury Hotel subforum that bookings on the hotels own programme rates will be prioritised for upgrades before Amex FHR bookings but my own experience (only done it once so far) was that I didn't receive an upgrade at the last Conrad I stayed at, but this might be because I already booked a suite and the next room category up is not upgradable.

As an example, I'm booked in for Conrad Tokyo March 2023 during cherry blossom season (fingers crossed Japan reopens to Pre-Covid entry requirements by then) and my travel agent's quoted price was about $1,500 AUD cheaper than the Amex FHR rate for a 5 night stay in a corner suite. Only difference between the benefit was that I get $100 USD hotel credit as opposed to $125 USD, and no late checkout. I thought it might have been a mistake and asked him to double check this but he confirmed it was the right rate that he was seeing. It's possible that there's been a pricing error but for $1,500 AUD i'll take that chance.

You can find TAs in the Luxury Hotel forum or I could refer mine to you if want.