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My guess would be no, because of the specifics of the case interface (it's not-quite-straight on the top and curved underneath) but I've not seen it tried. Maybe it could work.
Edit: confirmed that it doesn't fit.
I'm not looking to split though, sorry. Seiko will sell you one I'm sure, but I would guess it's not cheap since they actually put some effort into it and it's a proper five-link with solid end links made from two different machined parts joined together.
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Uncle Straps (formerly Uncle Seiko) make a number of bracelets for the SPB Willards though, including an engineer style one.
Their UK stockist (which I'm pretty sure is owned by them) doesn't have the engineer one though.
I’m fine with it (I have the white one and really like it). My preference with Seikos is generally for a nicely framed window at 3 but they so rarely if ever do that now.
Mine is for sale if you’re interested. I think I said £500 to someone on here. Has both the metal bracelet and the OEM rubber strap which I purchased separately, boxed, spare links etc.