should I rent a car in Latvia?

It is always good to read the UK Gov Foreign travel advice for a country (link to Latvia).

Check Safety and security --> Belarus border, Crime and Transport risks.

I am not sure if this applies to you: https://www.rs.gov.lv/en/borderland-regime

Maybe pictures of Latvia (Lettland) from German Federal Archive help you in any way? If you want to search other stuff use German names of places.
https://www.bild.bundesarchiv.de/dba/en/search/?yearfrom=&yearto=&query=Lettland

I also recommend reading the German Wikipedia page of Ludza which contains more details of the part of history you seem to be interested in, it contains also a today picture for the former synagogue.

DeepL Translator delivers best translation quality from / to German.

I do not know when your ancestors settled over but a lot of them used Hamburg and Bremen (in Gemany) as ports of emigration. There are still archives, e. g. Ballinstadt in Hamburg.

I have not been to eastern Latvia yet, but we did a road trip through all three Baltic states last summer with a rental. It was actually not that expensive to start in one country and drop off in another (in fact, we were doing the transfer back for the rental company). So you could also check another rental company.

Driving was smooth, and completely unproblematic - of course, only a personal experience, but still.

I understand the problem with your mum - if it were rational fear, it could probably be tackled with facts, but that kind of fear is often obscure and irrational, and therefore hard to solve. My in-laws had similar concerns when they heard we were driving through Canada (we are German), but they decided that it is probably best if they do not know.
You can of course try with rational facts, if you feel this helps - crime rates are low in the Baltics even for European standards, which in general are (sometimes significantly) lower than in the US. But you might have already tried that - and if so, it apparently did not help.
Just to provide a little context: You seem to live in Detroit, which in German media is only covered as some failed state equivalent of crime ridden lawlessness (if at all), just one step above Somalia. So if she is cool with that, the Baltics are a piece of cake.

I would say that Rundale Palace is definitely worth a stop. It reminded me of a small Versailles. You might consider a day trip to Rundale - there are several offered from Riga that also include Bauskas.

For driving: I would say to save your day of car rental for your trip to Ludza where you don’t have other good options. OR email your lodging and see if they can help you with a driver for the day - maybe the same (or less) cost and lots less hassle.

You already know that your mom’s fears are not rational nor will they probably change. You are already doing something she will freak out about (the whole trip) so you need to make the best decision for yourself. While driving in Latvia wasn’t hard, I drive all the time in lots of places. It sounds like you do not regularly drive, so a tour or a driver might be a better choice. Don’t drive IN Riga.

Ludza is the reason for the trip, so that is one place it makes sense to not skimp.

My apartment in Michigan is actually in a suburb just north of Detroit, not technically in the city of Detroit. I have driven solo to the Detroit Institute of Art, Belle Isle, and Riverwalk in Detroit. Some parts of Detroit are ok to visit as a tourist. Most of the crime is in the gang-infested ghetto residential neighborhoods the average tourist has no reason to go to. My mom grew up in the ghetto in North-west Detroit. I own a car and drive in Michigan like a "normal person". I drive just under an average of 7 thousand miles a year while some opinion say the average driver drives twelve thousand miles a year. Typically I don't go to events or places that would take more than 45 minutes driving unless it is on a weekend and I am quite sure the place or event would be worth it. I think 5 times before renting a car in travel far from home. I have driven to Chicago, Ill, Lake Michigan, the northern tip on Lower MI, and Niagara Falls NY. I don't have an anxiety about the action of actual driving. I only have low-level anxiety I probably can sublimate ok, about feeling guilty for "doing something dangerous" which might or might not be related to my mom's reactions or genetic accident. Driving is just transportation to me. It isn't my favorite activity. I have irrationally driven to places I didn't need to go to. ... I don't have a fear of driving or getting lost. I am not sure whether I "should" be afraid to rent a car due to the possibility of carjacking or getting shot while driving. Not finding a parking spot would be my top non-irrational worry. In Miami Fl and Southern CA I rented a car but the parking situation was too tight and I wish I had just used taxis or Uber or Lyft.... My great-grandfather and his wife came to the USA in 1914 so he could avoid serving in the Russian army in World War One. ... Ludza is more than two miles from the Russian border. Check the map...

Supposed rest of my itinerary:

Friday various museums in Kaunas. 8th night, 3rd night in Kaunas
Saturday: Bus to Riga. 9th night. 1st night in Riga
Sunday: day trip with a rental car to Rundale palace and Bauskas Castle. 10th night. 2nd night in Riga
Monday: Various museums in Riga. 11th night. 3rd night in Riga
Tuesday: Taxi or rideshare to pick up a rental car. Drive to Ludza. Find the grave, whatever else if time. 12th night. 1st night in Ludza.
Wednesday: Local history museum, castle ruins, site of mass shooting, lakes, various streets. 13th night. 2nd night in Ludza.
Thursday: Drive back to Riga, return rental car, taxi or rideshare to hotel. 14th night. 1st night of second stay in Riga.
Friday: Various museums. 15th night. 2nd night of second stay in Riga
Saturday: Various museums. 16th night. 3rd night of second stay in Riga
Sunday: Flights back to Detroit leave at 6:50am

Do you think my itinerary has too much time in Vilnius and not enough time in Riga?

I personally found Riga better than Vilnius - but that is a matter of taste, so hard to assess. I might have gotten lost, but how many nights was it in each place?

I had 2 full days + arrival days by 12 pm in both Riga and Vilnius--with no day trips from either city, and a good plan for what I wanted to see--and ended up seeing more than that--I got a good feeling for each. More time, another full day, perhaps, would have been great, but easy to make good use of my time as each was quite walkable. If I had to choose more time in one over the other, it would be Riga. Why? I was there over a holiday, their reclamation of Independence Day--lots to see and enjoy, no complaints--and thus missed a museum and a vintage flea market, plus I am a knitter and enjoyed their shops devoted to their unique knitting traditions...and the food was fabulous!