Advice: friend too heavy for horse

I would tell her exactly like you told us. That you don't have the other horses available no more, that your horse has the clubfoot and a history of back problems.
Can she be of help to you when you work? You could offer her this, it might be just enough to make her feel better. And who knows, she might even consider dieting as her own idea

I'd tell her simply that she is aging and she can't carry what she used to. You really don't want to put her health in jeopardy. I agree, fat people know there fat. She will understand, although she will probably feel bad. I'd try to give her other options like riding a different horse, even if she has a pay to rent it for a day.
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Speaking as a big person:

Don't lie or make excuses, tell the truth. There should be nothing worse in telling a friend that she is not a good match for your horse because of her size, than there is saying it because of the fact that she isn't experienced enough, or any other reason.

It is far easier to be honest and matter of fact, than to try and not hurt her feelings. Again speaking as a bigger person, it is no shock that I am fat, I do know it, I don't want you to judge me, but facts are facts, if I picked up a size 10 pair of jeans you would tell me that they wouldn't fit, same as your horse, not the right fit at the moment.

Oops double post

Ya think? I have shared with her all of these details, about the club foot, the lameness years ago, the back soreness, she knows how valuable the horse is to my livelihood. I just haven't suggested that she might weigh too much. I suppose I could offer for her to help. She says she has been dieting for quite some time now. She has cut all sugar out of her diet, no caffeine, no drugs, doesn't ever drink.

Say her condition has worsened and you really need to be selective of who you put on her. Then tell her she may not ride her any more. Simple as that.
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Having the criterion be must be able to mount from the ground is not a good idea. I cannot mount from the ground, but that is due to bad knees. And, I would not want a heavy person trying, over and over, struggling and pulling on my horse's back . That's much worse than having them mount politely from a block.

If she is really attached to the horse you can also offer to let her spend time with the horse on the ground... Barring the horse is being used of course.
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Why don't you tell her the truth, then support her weight loss? Maybe you can help her work-out in lieu of lessons, make the weight-loss horse themed with lunges, maybe she can jog alongside the horse, be outdoors working out, running wheelbarrows, carrying buckets, cleaning stalls, and the goal is if she loses x amount of pounds, she can ride and it will be a celebratory event! They did it at my old barn. The instructor would have an equine fitness class for her students 2x a week. They would lay out in the pasture with their yoga mats and work out on exercises for their wellbeing and for helping them ride better.

This doesn't change the fact that right now she's too big for the horse. I know you want to be supportive. I have a friend who just recently lost 200 lbs. She used to ride horses and at her heaviest she asked me to ride my horse at the time. I told her I did not think it was a good idea and explained why. She was a little hurt but understood. The next week she asked me to start working out with her. Who knows, you might motivate your friend to lose some weight. Tell her if she drops so many pounds she can ride.