You definitely can have tomato plants large enough to plant out in five weeks if you start the seeds today. The ones I planted on 4/5/24 are six to 8 inches tall already and would be bigger if I had thinned and repotted them. That's just a little over three weeks.
A heat mat will speed up germination. You could try pre-soaking the seeds as well. After they germinate, put them outside in the sun on warm days (but be sure to acclimate them to the sun first so you don't burn them). They'll grow much faster in the sun than indoors.
Are you sure the ones you have aren't going to survive? If the stems broke partially but not all the way through, just repot the plants with the broken part under the soil. If the stems broke completely, you can cut off the top part of the plant above the break, or cut off any suckers, and root them in water or potting mix.
Other ideas: do you have a farmer's market in the area, where vendors might sell tomato plants? Often they'll have heirlooms not carried by the big box stores. Also check whatever local social media you have, like NextDoor or Facebook marketplace or Craigslist. Around here, we have a lot of individuals selling vegetable plants online this time of year. Also look for fundraiser plant sales by schools or Master Gardener-type organizations. Those may have less-common varieties.
I think I'd try a combination of rescuing some of the damaged seedlings, planting some new seeds of Inciardi & Andean to be sure to have them, and buying some larger transplants of whatever you can find so you get some tomatoes earlier in the season.
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