My 2005 4.4 had same problem but diagnostic software showed the culprit to be the alternator. It's air-cooled and yours is likely very much the same. It's not hard to replace. Take the opportunity to replace all the pulleys/tensioners and belts if they're not already replaced.

I got mine for $352.99 (with lifetime warranty and free shipping) from
2005 bmw x5 4.4l alternator Prices - 2005 bmw x5 4.4l alternators for Sale - Buy 2005 bmw x5 4.4l alternators



Here's my post for alternator http://www.xoutpost.com/bmw-sav-foru...005-model.html

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'05 E53 X5 4.4i, '97 E39 528, '07 E92 335i, '16 F86 X6M.