If by "the alliance", you mean the UK, USA, Australia & New Zealand, then buying from within it either means buying whatever the RAN buys (which has obvious advantages), regardless of its origins (outside '"the alliance", for recent ships) or Type 26, unless the USA comes up with something suitable for the RNZN, which currently seems unlikely. I don't see LCS as meeting your needs.
Citing the choice of the MEKO 200 for the ANZAC platform as a problem with European (non-UK) designs is perverse. It was too small, & still is, but whose fault is that? There were larger MEKO hulls available, but they weren't chosen. The Germans would also have been happy to sell a ship based on the significantly larger F123 Brandenburg hull, which was being built when the ANZACs were ordered, but that wasn't wanted. The too-small hull was the size specified by Australia & NZ, which meant that a scaled-down Type 23 had to be offered to meet it.
BTW, the ANZACs aren't 20% bigger than most MEKO frigates in service. 20% more displacement than some, i.e. the lightest.
NATO ships seem perfectly capable of working together, e.g. in the Gulf, & in operations off the former Yugoslavia, Libya, etc.