Most off the peg bridles seem to come with flash attachments these days so I get mine from here www.risleysaddlery.co.uk they sell all parts of the bridle separately so you can get the correct size of everything (I have bridles with horse headpiece, cob cheek pieces, cob nose band and horse browband for example) and whatever nose band you want in whatever thickness you want too.
I have three bridles from then and all are great quality. Wouldn't bother with off the peg now ever
Have a look at Henry James, they sell parts so you could just get a replacement head piece but a whole bridle will be more than you want to spend, they do them without a flash or removable flash.
For a whole bridle Shires do a nice range, the Aviemore is nice leather and a good price, you can get plain cavessons as well.
I’ve just bought a beautiful quality bridle from Henry James Saddlery. I bought just the headpiece, cheek pieces and noseband as my current browband was suitable to use. It was £80 but I did get it in the sale. I’m very pleased with it as I like the unusual design and it’s not bulky and weirdly feels lighter than a conventional bridle. It’s worth having a look.