You are probably as well getting the suspension setup done - the removal/fitting of components can be enough to knock it out (and it is not that expensive anyway).
I bought a used NSF lower wishbone for a MK1 from Autolink on Saturday. £34 including bushes, bolts and arb droplink. Bit of WD40 and the bolts will be fine I expect.
Google mx5 ball joint replacement or variations on that theme.
The rear inner bolt can be a pig because you cannot get enough leverage on the nut unless the car is on a ramp. I ended up taking off the head of the bolt with a powerfile. On my MX5 (1996, 128k miles) two of the bolts sheared, but they all tapped out easily with a drift. Make a note/digital photo of the bolt plate positions before you start, so that you can get them back close to the correct positions.
The simplest method for the inner (adjustment) bolts is to buy four new ones and get a friendly garage to replace them. Take photos of the bolts beforehand so that the new bolts can be aligned properly. When the car is on a lift it is easy to get a long bar breaker, put it on the head of the bolt and give it a good pull. The bolt will almost certainly snap and then just drift out. (Oddly these bolts do not rust in/seize). Without a lift you cannot get the clearance for a bar breaker. The alternative is to grind off the heads of the bolts, keeping the bushes cool by spraying them with water. I’ve done it - it takes bl**dy hours!